SOLVING THE SYSTEM OF EQUATIONS AND ADVANCED ALGEBRAIC
EXPRESSIONS
1. Solving the system of equations:
{x+y=10x−y=4\begin{} x + y = 10 \\ x - y = 4 \end{}{x+y=10x−y=4
Add the two equations:
(x+y)+(x−y)=10+42x=14x=7(x + y) + (x - y) = 10 + 4 \\ 2x = 14 \\ x = 7(x+y)+
(x−y)=10+42x=14x=7
Substitute x=7x = 7x=7 into the first equation:
7+y=10y=37 + y = 10 \\ y = 37+y=10y=3
So, x=7x = 7x=7 and y=3y = 3y=3.
2. Solving the system of equations:
{2a+3b=74a−b=5\begin{} 2a + 3b = 7 \\ 4a - b = 5 \end{}{2a+3b=74a−b=5
Multiply the second equation by 3 to eliminate bbb:
4a−b=512a−3b=154a - b = 5 \\ 12a - 3b = 154a−b=512a−3b=15
Add this to the first equation:
(2a+3b)+(12a−3b)=7+1514a=22a=2214=117(2a + 3b) + (12a - 3b) = 7 + 15 \\ 14a = 22 \\ a
= \{22}{14} = \{11}{7}(2a+3b)+(12a−3b)=7+1514a=22a=1422=711
Substitute a=117a = \{11}{7}a=711 into the first equation:
2(117)+3b=7227+3b=73b=7−2273b=497−2273b=277b=2721=972\left(\{11}{7}\) + 3b = 7 \\
\{22}{7} + 3b = 7 \\ 3b = 7 - \{22}{7} \\ 3b = \{49}{7} - \{22}{7} \\ 3b = \{27}{7} \\ b = \
{27}{21} = \{9}{7}2(711)+3b=7722+3b=73b=7−7223b=749−7223b=727b=2127=79
So, a=117a = \{11}{7}a=711 and b=97b = \{9}{7}b=79.
3. Solving the equation:
x+23+x−12=4\{x+2}{3} + \{x-1}{2} = 43x+2+2x−1=4
Find a common denominator (6):
2(x+2)6+3(x−1)6=42x+4+3x−36=45x+16=45x+1=245x=23x=235=4.6\{2(x+2)}{6} + \{3(x-
1)}{6} = 4 \\ \{2x + 4 + 3x - 3}{6} = 4 \\ \{5x + 1}{6} = 4 \\ 5x + 1 = 24 \\ 5x = 23 \\ x = \
{23}{5} = 4.662(x+2)+63(x−1)=462x+4+3x−3=465x+1=45x+1=245x=23x=523=4.6
4. Solving the equation:
2y+35−y−24=1\{2y+3}{5} - \{y-2}{4} = 152y+3−4y−2=1
Find a common denominator (20):
4(2y+3)20−5(y−2)20=18y+12−5y+1020=13y+2220=13y+22=203y=−2y=−23\{4(2y+3)}
{20} - \{5(y-2)}{20} = 1 \\ \{8y + 12 - 5y + 10}{20} = 1 \\ \{3y + 22}{20} = 1 \\ 3y + 22 =
20 \\ 3y = -2 \\ y = -\{2}{3}204(2y+3)−205(y−2)=1208y+12−5y+10=1203y+22
=13y+22=203y=−2y=−32
5. Solving the equation:
3z−14+z+23=5\{3z-1}{4} + \{z+2}{3} = 543z−1+3z+2=5
Find a common denominator (12):
3(3z−1)12+4(z+2)12=59z−3+4z+812=513z+512=513z+5=6013z=55z=5513≈4.23\{3(3z-1)}
{12} + \{4(z+2)}{12} = 5 \\ \{9z - 3 + 4z + 8}{12} = 5 \\ \{13z + 5}{12} = 5 \\ 13z + 5 =
60 \\ 13z = 55 \\ z = \{55}{13} \approx 4.23123(3z−1)+124(z+2)=5129z−3+4z+8=51213z+5
=513z+5=6013z=55z=1355≈4.23
6. Solving the equation:
2(a−3)+3(a+4)=52(a-3) + 3(a+4) = 52(a−3)+3(a+4)=5
Expand and combine like terms:
2a−6+3a+12=55a+6=55a=−1a=−152a - 6 + 3a + 12 = 5 \\ 5a + 6 = 5 \\ 5a = -1 \\ a = -\{1}
{5}2a−6+3a+12=55a+6=55a=−1a=−51
7. Solving the equation:
5(b+2)−2(b−3)=105(b+2) - 2(b-3) = 105(b+2)−2(b−3)=10
Expand and combine like terms:
5b+10−2b+6=103b+16=103b=−6b=−25b + 10 - 2b + 6 = 10 \\ 3b + 16 = 10 \\ 3b = -6 \\ b = -
25b+10−2b+6=103b+16=103b=−6b=−2
8. Solving the equation:
(x+1)2−4(x−2)=0(x+1)^2 - 4(x-2) = 0(x+1)2−4(x−2)=0
Expand and simplify:
(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0(x+1)^2 - 4x + 8 = 0 \\ x^2 + 2x + 1 - 4x + 8 =
0 \\ x^2 - 2x + 9 = 0(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0
Solve using the quadratic formula x=−b±b2−4ac2ax = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}x=2a−b±b2−4ac:
x=2±4−362x=2±−322x=1±i8x = \{2 \pm \sqrt{4 - 36}}{2} \\ x = \{2 \pm \sqrt{-32}}{2} \\ x
= 1 \pm i\sqrt{8}x=22±4−36x=22±−32x=1±i8
So, x=1+22ix = 1 + 2\sqrt{2}ix=1+22i and x=1−22ix = 1 - 2\sqrt{2}ix=1−22i.
5. Solving the equation:
3z−14+z+23=5\{3z-1}{4} + \{z+2}{3} = 543z−1+3z+2=5
Find a common denominator (12):
3(3z−1)12+4(z+2)12=59z−3+4z+812=513z+512=513z+5=6013z=55z=5513≈4.23\{3(3z-1)}
{12} + \{4(z+2)}{12} = 5 \\ \{9z - 3 + 4z + 8}{12} = 5 \\ \{13z + 5}{12} = 5 \\ 13z + 5 =
60 \\ 13z = 55 \\ z = \{55}{13} \approx 4.23123(3z−1)+124(z+2)=5129z−3+4z+8=51213z+5
=513z+5=6013z=55z=1355≈4.23
6. Solving the equation:
2(a−3)+3(a+4)=52(a-3) + 3(a+4) = 52(a−3)+3(a+4)=5
Expand and combine like terms:
2a−6+3a+12=55a+6=55a=−1a=−152a - 6 + 3a + 12 = 5 \\ 5a + 6 = 5 \\ 5a = -1 \\ a = -\{1}
{5}2a−6+3a+12=55a+6=55a=−1a=−51
7. Solving the equation:
5(b+2)−2(b−3)=105(b+2) - 2(b-3) = 105(b+2)−2(b−3)=10
Expand and combine like terms:
5b+10−2b+6=103b+16=103b=−6b=−25b + 10 - 2b + 6 = 10 \\ 3b + 16 = 10 \\ 3b = -6 \\ b = -
25b+10−2b+6=103b+16=103b=−6b=−2
8. Solving the equation:
(x+1)2−4(x−2)=0(x+1)^2 - 4(x-2) = 0(x+1)2−4(x−2)=0
Expand and simplify:
(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0(x+1)^2 - 4x + 8 = 0 \\ x^2 + 2x + 1 - 4x + 8 =
0 \\ x^2 - 2x + 9 = 0(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0
Solve using the quadratic formula x=−b±b2−4ac2ax = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}x=2a−b±b2−4ac:
x=2±4−362x=2±−322x=1±i8x = \{2 \pm \sqrt{4 - 36}}{2} \\ x = \{2 \pm \sqrt{-32}}{2} \\ x
= 1 \pm i\sqrt{8}x=22±4−36x=22±−32x=1±i8
So, x=1+22ix = 1 + 2\sqrt{2}ix=1+22i and x=1−22ix = 1 - 2\sqrt{2}ix=1−22i.
9. Solving the equation:
3(y−1)2+2(y+3)=03(y-1)^2 + 2(y+3) = 03(y−1)2+2(y+3)=0
Expand and combine like terms:
3(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=03(y^2 - 2y + 1) + 2y + 6 = 0 \\ 3y^2 - 6y
+ 3 + 2y + 6 = 0 \\ 3y^2 - 4y + 9 = 03(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=0
Solve using the quadratic formula y=−b±b2−4ac2ay = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}y=2a−b±b2−4ac:
y=4±16−1086y=4±−926y=4±2i236y=2±i233y = \{4 \pm \sqrt{16 - 108}}{6} \\ y = \{4 \pm \
sqrt{-92}}{6} \\ y = \{4 \pm 2i\sqrt{23}}{6} \\ y = \{2 \pm i\sqrt{23}}{3}y=64±16−108
y=64±−92y=64±2i23y=32±i23
So, y=2+i233y = \{2 + i\sqrt{23}}{3}y=32+i23 and y=2−i233y = \{2 - i\sqrt{23}}
{3}y=32−i23.
10. Solving the equation:
(z−2)2−5(z+1)=0(z-2)^2 - 5(z+1) = 0(z−2)2−5(z+1)=0
Expand and simplify:
z2−4z+4−5z−5=0z2−9z−1=0z^2 - 4z + 4 - 5z - 5 = 0 \\ z^2 - 9z - 1 =
0z2−4z+4−5z−5=0z2−9z−1=0
Solve using the quadratic formula z=−b±b2−4ac2az = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}z=2a−b±b2−4ac:
z=9±81+42z=9±852z = \{9 \pm \sqrt{81 + 4}}{2} \\ z = \{9 \pm \sqrt{85}}{2}z=29±81+4
z=29±85
So, z=9+852z = \{9 + \sqrt{85}}{2}z=29+85 and z=9−852z = \{9 - \sqrt{85}}{2}z=29−85.
Advanced Algebraic Expressions
61. Simplify:
3x2−9x3x\frac{3x^2 - 9x}{3x}3x3x2−9x
Factor the numerator:
3x(x−3)3x=x−3\frac{3x(x - 3)}{3x} = x - 33x3x(x−3)=x−3
62. Simplify:
4y2−16y4y\frac{4y^2 - 16y}{4y}4y4y2−16y
Factor the numerator:
4y(y−4)4y=y−4\frac{4y(y - 4)}{4y} = y - 44y4y(y−4)=y−4
63. Simplify:
2z2+8z2z\frac{2z^2 + 8z}{2z}2z2z2+8z
Factor the numerator:
2z(z+4)2z=z+4\frac{2z(z + 4)}{2z} = z + 42z2z(z+4)=z+4
64. Expand:
(2x−3)2(2x - 3)^2(2x−3)2
Use the binomial expansion formula (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab +
b^2(a−b)2=a2−2ab+b2:
(2x−3)2=4x2−12x+9(2x - 3)^2 = 4x^2 - 12x + 9(2x−3)2=4x2−12x+9
65. Expand:
(y+4)2(y + 4)^2(y+4)2
Use the binomial expansion formula (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab +
b^2(a+b)2=a2+2ab+b2:
(y+4)2=y2+8y+16(y + 4)^2 = y^2 + 8y + 16(y+4)2=y2+8y+16
66. Expand:
(3a−2b)(3a+2b)(3a - 2b)(3a + 2b)(3a−2b)(3a+2b)
Use the difference of squares formula (a−b)(a+b)=a2−b2(a - b)(a + b) = a^2 - b^2(a−b)
(a+b)=a2−b2:
(3a−2b)(3a+2b)=9a2−4b2(3a - 2b)(3a + 2b) = 9a^2 - 4b^2(3a−2b)(3a+2b)=9a2−4b2
67. Factor:
x2−9x^2 - 9x2−9
This is a difference of squares:
5. Solving the equation:
3z−14+z+23=5\{3z-1}{4} + \{z+2}{3} = 543z−1+3z+2=5
Find a common denominator (12):
3(3z−1)12+4(z+2)12=59z−3+4z+812=513z+512=513z+5=6013z=55z=5513≈4.23\{3(3z-1)}
{12} + \{4(z+2)}{12} = 5 \\ \{9z - 3 + 4z + 8}{12} = 5 \\ \{13z + 5}{12} = 5 \\ 13z + 5 =
60 \\ 13z = 55 \\ z = \{55}{13} \approx 4.23123(3z−1)+124(z+2)=5129z−3+4z+8=51213z+5
=513z+5=6013z=55z=1355≈4.23
6. Solving the equation:
2(a−3)+3(a+4)=52(a-3) + 3(a+4) = 52(a−3)+3(a+4)=5
Expand and combine like terms:
2a−6+3a+12=55a+6=55a=−1a=−152a - 6 + 3a + 12 = 5 \\ 5a + 6 = 5 \\ 5a = -1 \\ a = -\{1}
{5}2a−6+3a+12=55a+6=55a=−1a=−51
7. Solving the equation:
5(b+2)−2(b−3)=105(b+2) - 2(b-3) = 105(b+2)−2(b−3)=10
Expand and combine like terms:
5b+10−2b+6=103b+16=103b=−6b=−25b + 10 - 2b + 6 = 10 \\ 3b + 16 = 10 \\ 3b = -6 \\ b = -
25b+10−2b+6=103b+16=103b=−6b=−2
8. Solving the equation:
(x+1)2−4(x−2)=0(x+1)^2 - 4(x-2) = 0(x+1)2−4(x−2)=0
Expand and simplify:
(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0(x+1)^2 - 4x + 8 = 0 \\ x^2 + 2x + 1 - 4x + 8 =
0 \\ x^2 - 2x + 9 = 0(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0
Solve using the quadratic formula x=−b±b2−4ac2ax = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}x=2a−b±b2−4ac:
x=2±4−362x=2±−322x=1±i8x = \{2 \pm \sqrt{4 - 36}}{2} \\ x = \{2 \pm \sqrt{-32}}{2} \\ x
= 1 \pm i\sqrt{8}x=22±4−36x=22±−32x=1±i8
So, x=1+22ix = 1 + 2\sqrt{2}ix=1+22i and x=1−22ix = 1 - 2\sqrt{2}ix=1−22i.
9. Solving the equation:
3(y−1)2+2(y+3)=03(y-1)^2 + 2(y+3) = 03(y−1)2+2(y+3)=0
Expand and combine like terms:
3(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=03(y^2 - 2y + 1) + 2y + 6 = 0 \\ 3y^2 - 6y
+ 3 + 2y + 6 = 0 \\ 3y^2 - 4y + 9 = 03(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=0
Solve using the quadratic formula y=−b±b2−4ac2ay = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}y=2a−b±b2−4ac:
y=4±16−1086y=4±−926y=4±2i236y=2±i233y = \{4 \pm \sqrt{16 - 108}}{6} \\ y = \{4 \pm \
sqrt{-92}}{6} \\ y = \{4 \pm 2i\sqrt{23}}{6} \\ y = \{2 \pm i\sqrt{23}}{3}y=64±16−108
y=64±−92y=64±2i23y=32±i23
So, y=2+i233y = \{2 + i\sqrt{23}}{3}y=32+i23 and y=2−i233y = \{2 - i\sqrt{23}}
{3}y=32−i23.
10. Solving the equation:
(z−2)2−5(z+1)=0(z-2)^2 - 5(z+1) = 0(z−2)2−5(z+1)=0
Expand and simplify:
z2−4z+4−5z−5=0z2−9z−1=0z^2 - 4z + 4 - 5z - 5 = 0 \\ z^2 - 9z - 1 =
0z2−4z+4−5z−5=0z2−9z−1=0
Solve using the quadratic formula z=−b±b2−4ac2az = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}z=2a−b±b2−4ac:
z=9±81+42z=9±852z = \{9 \pm \sqrt{81 + 4}}{2} \\ z = \{9 \pm \sqrt{85}}{2}z=29±81+4
z=29±85
So, z=9+852z = \{9 + \sqrt{85}}{2}z=29+85 and z=9−852z = \{9 - \sqrt{85}}{2}z=29−85.
Advanced Algebraic Expressions
61. Simplify:
3x2−9x3x\frac{3x^2 - 9x}{3x}3x3x2−9x
Factor the numerator:
3x(x−3)3x=x−3\frac{3x(x - 3)}{3x} = x - 33x3x(x−3)=x−3
62. Simplify:
4y2−16y4y\frac{4y^2 - 16y}{4y}4y4y2−16y
Factor the numerator:
4y(y−4)4y=y−4\frac{4y(y - 4)}{4y} = y - 44y4y(y−4)=y−4
63. Simplify:
2z2+8z2z\frac{2z^2 + 8z}{2z}2z2z2+8z
Factor the numerator:
2z(z+4)2z=z+4\frac{2z(z + 4)}{2z} = z + 42z2z(z+4)=z+4
64. Expand:
(2x−3)2(2x - 3)^2(2x−3)2
Use the binomial expansion formula (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab +
b^2(a−b)2=a2−2ab+b2:
(2x−3)2=4x2−12x+9(2x - 3)^2 = 4x^2 - 12x + 9(2x−3)2=4x2−12x+9
65. Expand:
(y+4)2(y + 4)^2(y+4)2
Use the binomial expansion formula (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab +
b^2(a+b)2=a2+2ab+b2:
(y+4)2=y2+8y+16(y + 4)^2 = y^2 + 8y + 16(y+4)2=y2+8y+16
66. Expand:
(3a−2b)(3a+2b)(3a - 2b)(3a + 2b)(3a−2b)(3a+2b)
Use the difference of squares formula (a−b)(a+b)=a2−b2(a - b)(a + b) = a^2 - b^2(a−b)
(a+b)=a2−b2:
(3a−2b)(3a+2b)=9a2−4b2(3a - 2b)(3a + 2b) = 9a^2 - 4b^2(3a−2b)(3a+2b)=9a2−4b2
67. Factor:
x2−9x^2 - 9x2−9
This is a difference of squares:
5. Solving the equation:
3z−14+z+23=5\{3z-1}{4} + \{z+2}{3} = 543z−1+3z+2=5
Find a common denominator (12):
3(3z−1)12+4(z+2)12=59z−3+4z+812=513z+512=513z+5=6013z=55z=5513≈4.23\{3(3z-1)}
{12} + \{4(z+2)}{12} = 5 \\ \{9z - 3 + 4z + 8}{12} = 5 \\ \{13z + 5}{12} = 5 \\ 13z + 5 =
60 \\ 13z = 55 \\ z = \{55}{13} \approx 4.23123(3z−1)+124(z+2)=5129z−3+4z+8=51213z+5
=513z+5=6013z=55z=1355≈4.23
6. Solving the equation:
2(a−3)+3(a+4)=52(a-3) + 3(a+4) = 52(a−3)+3(a+4)=5
Expand and combine like terms:
2a−6+3a+12=55a+6=55a=−1a=−152a - 6 + 3a + 12 = 5 \\ 5a + 6 = 5 \\ 5a = -1 \\ a = -\{1}
{5}2a−6+3a+12=55a+6=55a=−1a=−51
7. Solving the equation:
5(b+2)−2(b−3)=105(b+2) - 2(b-3) = 105(b+2)−2(b−3)=10
Expand and combine like terms:
5b+10−2b+6=103b+16=103b=−6b=−25b + 10 - 2b + 6 = 10 \\ 3b + 16 = 10 \\ 3b = -6 \\ b = -
25b+10−2b+6=103b+16=103b=−6b=−2
8. Solving the equation:
(x+1)2−4(x−2)=0(x+1)^2 - 4(x-2) = 0(x+1)2−4(x−2)=0
Expand and simplify:
(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0(x+1)^2 - 4x + 8 = 0 \\ x^2 + 2x + 1 - 4x + 8 =
0 \\ x^2 - 2x + 9 = 0(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0
Solve using the quadratic formula x=−b±b2−4ac2ax = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}x=2a−b±b2−4ac:
x=2±4−362x=2±−322x=1±i8x = \{2 \pm \sqrt{4 - 36}}{2} \\ x = \{2 \pm \sqrt{-32}}{2} \\ x
= 1 \pm i\sqrt{8}x=22±4−36x=22±−32x=1±i8
So, x=1+22ix = 1 + 2\sqrt{2}ix=1+22i and x=1−22ix = 1 - 2\sqrt{2}ix=1−22i.
9. Solving the equation:
3(y−1)2+2(y+3)=03(y-1)^2 + 2(y+3) = 03(y−1)2+2(y+3)=0
Expand and combine like terms:
3(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=03(y^2 - 2y + 1) + 2y + 6 = 0 \\ 3y^2 - 6y
+ 3 + 2y + 6 = 0 \\ 3y^2 - 4y + 9 = 03(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=0
Solve using the quadratic formula y=−b±b2−4ac2ay = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}y=2a−b±b2−4ac:
y=4±16−1086y=4±−926y=4±2i236y=2±i233y = \{4 \pm \sqrt{16 - 108}}{6} \\ y = \{4 \pm \
sqrt{-92}}{6} \\ y = \{4 \pm 2i\sqrt{23}}{6} \\ y = \{2 \pm i\sqrt{23}}{3}y=64±16−108
y=64±−92y=64±2i23y=32±i23
So, y=2+i233y = \{2 + i\sqrt{23}}{3}y=32+i23 and y=2−i233y = \{2 - i\sqrt{23}}
{3}y=32−i23.
10. Solving the equation:
(z−2)2−5(z+1)=0(z-2)^2 - 5(z+1) = 0(z−2)2−5(z+1)=0
Expand and simplify:
z2−4z+4−5z−5=0z2−9z−1=0z^2 - 4z + 4 - 5z - 5 = 0 \\ z^2 - 9z - 1 =
0z2−4z+4−5z−5=0z2−9z−1=0
Solve using the quadratic formula z=−b±b2−4ac2az = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}z=2a−b±b2−4ac:
z=9±81+42z=9±852z = \{9 \pm \sqrt{81 + 4}}{2} \\ z = \{9 \pm \sqrt{85}}{2}z=29±81+4
z=29±85
So, z=9+852z = \{9 + \sqrt{85}}{2}z=29+85 and z=9−852z = \{9 - \sqrt{85}}{2}z=29−85.
Advanced Algebraic Expressions
61. Simplify:
3x2−9x3x\frac{3x^2 - 9x}{3x}3x3x2−9x
Factor the numerator:
3x(x−3)3x=x−3\frac{3x(x - 3)}{3x} = x - 33x3x(x−3)=x−3
62. Simplify:
4y2−16y4y\frac{4y^2 - 16y}{4y}4y4y2−16y
Factor the numerator:
4y(y−4)4y=y−4\frac{4y(y - 4)}{4y} = y - 44y4y(y−4)=y−4
63. Simplify:
2z2+8z2z\frac{2z^2 + 8z}{2z}2z2z2+8z
Factor the numerator:
2z(z+4)2z=z+4\frac{2z(z + 4)}{2z} = z + 42z2z(z+4)=z+4
64. Expand:
(2x−3)2(2x - 3)^2(2x−3)2
Use the binomial expansion formula (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab +
b^2(a−b)2=a2−2ab+b2:
(2x−3)2=4x2−12x+9(2x - 3)^2 = 4x^2 - 12x + 9(2x−3)2=4x2−12x+9
65. Expand:
(y+4)2(y + 4)^2(y+4)2
Use the binomial expansion formula (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab +
b^2(a+b)2=a2+2ab+b2:
(y+4)2=y2+8y+16(y + 4)^2 = y^2 + 8y + 16(y+4)2=y2+8y+16
66. Expand:
(3a−2b)(3a+2b)(3a - 2b)(3a + 2b)(3a−2b)(3a+2b)
Use the difference of squares formula (a−b)(a+b)=a2−b2(a - b)(a + b) = a^2 - b^2(a−b)
(a+b)=a2−b2:
(3a−2b)(3a+2b)=9a2−4b2(3a - 2b)(3a + 2b) = 9a^2 - 4b^2(3a−2b)(3a+2b)=9a2−4b2
67. Factor:
x2−9x^2 - 9x2−9
This is a difference of squares:
5. Solving the equation:
3z−14+z+23=5\{3z-1}{4} + \{z+2}{3} = 543z−1+3z+2=5
Find a common denominator (12):
3(3z−1)12+4(z+2)12=59z−3+4z+812=513z+512=513z+5=6013z=55z=5513≈4.23\{3(3z-1)}
{12} + \{4(z+2)}{12} = 5 \\ \{9z - 3 + 4z + 8}{12} = 5 \\ \{13z + 5}{12} = 5 \\ 13z + 5 =
60 \\ 13z = 55 \\ z = \{55}{13} \approx 4.23123(3z−1)+124(z+2)=5129z−3+4z+8=51213z+5
=513z+5=6013z=55z=1355≈4.23
6. Solving the equation:
2(a−3)+3(a+4)=52(a-3) + 3(a+4) = 52(a−3)+3(a+4)=5
Expand and combine like terms:
2a−6+3a+12=55a+6=55a=−1a=−152a - 6 + 3a + 12 = 5 \\ 5a + 6 = 5 \\ 5a = -1 \\ a = -\{1}
{5}2a−6+3a+12=55a+6=55a=−1a=−51
7. Solving the equation:
5(b+2)−2(b−3)=105(b+2) - 2(b-3) = 105(b+2)−2(b−3)=10
Expand and combine like terms:
5b+10−2b+6=103b+16=103b=−6b=−25b + 10 - 2b + 6 = 10 \\ 3b + 16 = 10 \\ 3b = -6 \\ b = -
25b+10−2b+6=103b+16=103b=−6b=−2
8. Solving the equation:
(x+1)2−4(x−2)=0(x+1)^2 - 4(x-2) = 0(x+1)2−4(x−2)=0
Expand and simplify:
(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0(x+1)^2 - 4x + 8 = 0 \\ x^2 + 2x + 1 - 4x + 8 =
0 \\ x^2 - 2x + 9 = 0(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0
Solve using the quadratic formula x=−b±b2−4ac2ax = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}x=2a−b±b2−4ac:
x=2±4−362x=2±−322x=1±i8x = \{2 \pm \sqrt{4 - 36}}{2} \\ x = \{2 \pm \sqrt{-32}}{2} \\ x
= 1 \pm i\sqrt{8}x=22±4−36x=22±−32x=1±i8
So, x=1+22ix = 1 + 2\sqrt{2}ix=1+22i and x=1−22ix = 1 - 2\sqrt{2}ix=1−22i.
9. Solving the equation:
3(y−1)2+2(y+3)=03(y-1)^2 + 2(y+3) = 03(y−1)2+2(y+3)=0
Expand and combine like terms:
3(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=03(y^2 - 2y + 1) + 2y + 6 = 0 \\ 3y^2 - 6y
+ 3 + 2y + 6 = 0 \\ 3y^2 - 4y + 9 = 03(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=0
Solve using the quadratic formula y=−b±b2−4ac2ay = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}y=2a−b±b2−4ac:
y=4±16−1086y=4±−926y=4±2i236y=2±i233y = \{4 \pm \sqrt{16 - 108}}{6} \\ y = \{4 \pm \
sqrt{-92}}{6} \\ y = \{4 \pm 2i\sqrt{23}}{6} \\ y = \{2 \pm i\sqrt{23}}{3}y=64±16−108
y=64±−92y=64±2i23y=32±i23
So, y=2+i233y = \{2 + i\sqrt{23}}{3}y=32+i23 and y=2−i233y = \{2 - i\sqrt{23}}
{3}y=32−i23.
10. Solving the equation:
(z−2)2−5(z+1)=0(z-2)^2 - 5(z+1) = 0(z−2)2−5(z+1)=0
Expand and simplify:
z2−4z+4−5z−5=0z2−9z−1=0z^2 - 4z + 4 - 5z - 5 = 0 \\ z^2 - 9z - 1 =
0z2−4z+4−5z−5=0z2−9z−1=0
Solve using the quadratic formula z=−b±b2−4ac2az = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}z=2a−b±b2−4ac:
z=9±81+42z=9±852z = \{9 \pm \sqrt{81 + 4}}{2} \\ z = \{9 \pm \sqrt{85}}{2}z=29±81+4
z=29±85
So, z=9+852z = \{9 + \sqrt{85}}{2}z=29+85 and z=9−852z = \{9 - \sqrt{85}}{2}z=29−85.
Advanced Algebraic Expressions
61. Simplify:
3x2−9x3x\frac{3x^2 - 9x}{3x}3x3x2−9x
Factor the numerator:
3x(x−3)3x=x−3\frac{3x(x - 3)}{3x} = x - 33x3x(x−3)=x−3
62. Simplify:
4y2−16y4y\frac{4y^2 - 16y}{4y}4y4y2−16y
Factor the numerator:
4y(y−4)4y=y−4\frac{4y(y - 4)}{4y} = y - 44y4y(y−4)=y−4
63. Simplify:
2z2+8z2z\frac{2z^2 + 8z}{2z}2z2z2+8z
Factor the numerator:
2z(z+4)2z=z+4\frac{2z(z + 4)}{2z} = z + 42z2z(z+4)=z+4
64. Expand:
(2x−3)2(2x - 3)^2(2x−3)2
Use the binomial expansion formula (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab +
b^2(a−b)2=a2−2ab+b2:
(2x−3)2=4x2−12x+9(2x - 3)^2 = 4x^2 - 12x + 9(2x−3)2=4x2−12x+9
65. Expand:
(y+4)2(y + 4)^2(y+4)2
Use the binomial expansion formula (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab +
b^2(a+b)2=a2+2ab+b2:
(y+4)2=y2+8y+16(y + 4)^2 = y^2 + 8y + 16(y+4)2=y2+8y+16
66. Expand:
(3a−2b)(3a+2b)(3a - 2b)(3a + 2b)(3a−2b)(3a+2b)
Use the difference of squares formula (a−b)(a+b)=a2−b2(a - b)(a + b) = a^2 - b^2(a−b)
(a+b)=a2−b2:
(3a−2b)(3a+2b)=9a2−4b2(3a - 2b)(3a + 2b) = 9a^2 - 4b^2(3a−2b)(3a+2b)=9a2−4b2
67. Factor:
x2−9x^2 - 9x2−9
This is a difference of squares:
5. Solving the equation:
3z−14+z+23=5\{3z-1}{4} + \{z+2}{3} = 543z−1+3z+2=5
Find a common denominator (12):
3(3z−1)12+4(z+2)12=59z−3+4z+812=513z+512=513z+5=6013z=55z=5513≈4.23\{3(3z-1)}
{12} + \{4(z+2)}{12} = 5 \\ \{9z - 3 + 4z + 8}{12} = 5 \\ \{13z + 5}{12} = 5 \\ 13z + 5 =
60 \\ 13z = 55 \\ z = \{55}{13} \approx 4.23123(3z−1)+124(z+2)=5129z−3+4z+8=51213z+5
=513z+5=6013z=55z=1355≈4.23
6. Solving the equation:
2(a−3)+3(a+4)=52(a-3) + 3(a+4) = 52(a−3)+3(a+4)=5
Expand and combine like terms:
2a−6+3a+12=55a+6=55a=−1a=−152a - 6 + 3a + 12 = 5 \\ 5a + 6 = 5 \\ 5a = -1 \\ a = -\{1}
{5}2a−6+3a+12=55a+6=55a=−1a=−51
7. Solving the equation:
5(b+2)−2(b−3)=105(b+2) - 2(b-3) = 105(b+2)−2(b−3)=10
Expand and combine like terms:
5b+10−2b+6=103b+16=103b=−6b=−25b + 10 - 2b + 6 = 10 \\ 3b + 16 = 10 \\ 3b = -6 \\ b = -
25b+10−2b+6=103b+16=103b=−6b=−2
8. Solving the equation:
(x+1)2−4(x−2)=0(x+1)^2 - 4(x-2) = 0(x+1)2−4(x−2)=0
Expand and simplify:
(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0(x+1)^2 - 4x + 8 = 0 \\ x^2 + 2x + 1 - 4x + 8 =
0 \\ x^2 - 2x + 9 = 0(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0
Solve using the quadratic formula x=−b±b2−4ac2ax = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}x=2a−b±b2−4ac:
x=2±4−362x=2±−322x=1±i8x = \{2 \pm \sqrt{4 - 36}}{2} \\ x = \{2 \pm \sqrt{-32}}{2} \\ x
= 1 \pm i\sqrt{8}x=22±4−36x=22±−32x=1±i8
So, x=1+22ix = 1 + 2\sqrt{2}ix=1+22i and x=1−22ix = 1 - 2\sqrt{2}ix=1−22i.
9. Solving the equation:
3(y−1)2+2(y+3)=03(y-1)^2 + 2(y+3) = 03(y−1)2+2(y+3)=0
Expand and combine like terms:
3(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=03(y^2 - 2y + 1) + 2y + 6 = 0 \\ 3y^2 - 6y
+ 3 + 2y + 6 = 0 \\ 3y^2 - 4y + 9 = 03(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=0
Solve using the quadratic formula y=−b±b2−4ac2ay = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}y=2a−b±b2−4ac:
y=4±16−1086y=4±−926y=4±2i236y=2±i233y = \{4 \pm \sqrt{16 - 108}}{6} \\ y = \{4 \pm \
sqrt{-92}}{6} \\ y = \{4 \pm 2i\sqrt{23}}{6} \\ y = \{2 \pm i\sqrt{23}}{3}y=64±16−108
y=64±−92y=64±2i23y=32±i23
So, y=2+i233y = \{2 + i\sqrt{23}}{3}y=32+i23 and y=2−i233y = \{2 - i\sqrt{23}}
{3}y=32−i23.
10. Solving the equation:
(z−2)2−5(z+1)=0(z-2)^2 - 5(z+1) = 0(z−2)2−5(z+1)=0
Expand and simplify:
z2−4z+4−5z−5=0z2−9z−1=0z^2 - 4z + 4 - 5z - 5 = 0 \\ z^2 - 9z - 1 =
0z2−4z+4−5z−5=0z2−9z−1=0
Solve using the quadratic formula z=−b±b2−4ac2az = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}z=2a−b±b2−4ac:
z=9±81+42z=9±852z = \{9 \pm \sqrt{81 + 4}}{2} \\ z = \{9 \pm \sqrt{85}}{2}z=29±81+4
z=29±85
So, z=9+852z = \{9 + \sqrt{85}}{2}z=29+85 and z=9−852z = \{9 - \sqrt{85}}{2}z=29−85.
Advanced Algebraic Expressions
61. Simplify:
3x2−9x3x\frac{3x^2 - 9x}{3x}3x3x2−9x
Factor the numerator:
3x(x−3)3x=x−3\frac{3x(x - 3)}{3x} = x - 33x3x(x−3)=x−3
62. Simplify:
4y2−16y4y\frac{4y^2 - 16y}{4y}4y4y2−16y
Factor the numerator:
4y(y−4)4y=y−4\frac{4y(y - 4)}{4y} = y - 44y4y(y−4)=y−4
63. Simplify:
2z2+8z2z\frac{2z^2 + 8z}{2z}2z2z2+8z
Factor the numerator:
2z(z+4)2z=z+4\frac{2z(z + 4)}{2z} = z + 42z2z(z+4)=z+4
64. Expand:
(2x−3)2(2x - 3)^2(2x−3)2
Use the binomial expansion formula (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab +
b^2(a−b)2=a2−2ab+b2:
(2x−3)2=4x2−12x+9(2x - 3)^2 = 4x^2 - 12x + 9(2x−3)2=4x2−12x+9
65. Expand:
(y+4)2(y + 4)^2(y+4)2
Use the binomial expansion formula (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab +
b^2(a+b)2=a2+2ab+b2:
(y+4)2=y2+8y+16(y + 4)^2 = y^2 + 8y + 16(y+4)2=y2+8y+16
66. Expand:
(3a−2b)(3a+2b)(3a - 2b)(3a + 2b)(3a−2b)(3a+2b)
Use the difference of squares formula (a−b)(a+b)=a2−b2(a - b)(a + b) = a^2 - b^2(a−b)
(a+b)=a2−b2:
(3a−2b)(3a+2b)=9a2−4b2(3a - 2b)(3a + 2b) = 9a^2 - 4b^2(3a−2b)(3a+2b)=9a2−4b2
67. Factor:
x2−9x^2 - 9x2−9
This is a difference of squares:
5. Solving the equation:
3z−14+z+23=5\{3z-1}{4} + \{z+2}{3} = 543z−1+3z+2=5
Find a common denominator (12):
3(3z−1)12+4(z+2)12=59z−3+4z+812=513z+512=513z+5=6013z=55z=5513≈4.23\{3(3z-1)}
{12} + \{4(z+2)}{12} = 5 \\ \{9z - 3 + 4z + 8}{12} = 5 \\ \{13z + 5}{12} = 5 \\ 13z + 5 =
60 \\ 13z = 55 \\ z = \{55}{13} \approx 4.23123(3z−1)+124(z+2)=5129z−3+4z+8=51213z+5
=513z+5=6013z=55z=1355≈4.23
6. Solving the equation:
2(a−3)+3(a+4)=52(a-3) + 3(a+4) = 52(a−3)+3(a+4)=5
Expand and combine like terms:
2a−6+3a+12=55a+6=55a=−1a=−152a - 6 + 3a + 12 = 5 \\ 5a + 6 = 5 \\ 5a = -1 \\ a = -\{1}
{5}2a−6+3a+12=55a+6=55a=−1a=−51
7. Solving the equation:
5(b+2)−2(b−3)=105(b+2) - 2(b-3) = 105(b+2)−2(b−3)=10
Expand and combine like terms:
5b+10−2b+6=103b+16=103b=−6b=−25b + 10 - 2b + 6 = 10 \\ 3b + 16 = 10 \\ 3b = -6 \\ b = -
25b+10−2b+6=103b+16=103b=−6b=−2
8. Solving the equation:
(x+1)2−4(x−2)=0(x+1)^2 - 4(x-2) = 0(x+1)2−4(x−2)=0
Expand and simplify:
(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0(x+1)^2 - 4x + 8 = 0 \\ x^2 + 2x + 1 - 4x + 8 =
0 \\ x^2 - 2x + 9 = 0(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0
Solve using the quadratic formula x=−b±b2−4ac2ax = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}x=2a−b±b2−4ac:
x=2±4−362x=2±−322x=1±i8x = \{2 \pm \sqrt{4 - 36}}{2} \\ x = \{2 \pm \sqrt{-32}}{2} \\ x
= 1 \pm i\sqrt{8}x=22±4−36x=22±−32x=1±i8
So, x=1+22ix = 1 + 2\sqrt{2}ix=1+22i and x=1−22ix = 1 - 2\sqrt{2}ix=1−22i.
9. Solving the equation:
3(y−1)2+2(y+3)=03(y-1)^2 + 2(y+3) = 03(y−1)2+2(y+3)=0
Expand and combine like terms:
3(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=03(y^2 - 2y + 1) + 2y + 6 = 0 \\ 3y^2 - 6y
+ 3 + 2y + 6 = 0 \\ 3y^2 - 4y + 9 = 03(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=0
Solve using the quadratic formula y=−b±b2−4ac2ay = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}y=2a−b±b2−4ac:
y=4±16−1086y=4±−926y=4±2i236y=2±i233y = \{4 \pm \sqrt{16 - 108}}{6} \\ y = \{4 \pm \
sqrt{-92}}{6} \\ y = \{4 \pm 2i\sqrt{23}}{6} \\ y = \{2 \pm i\sqrt{23}}{3}y=64±16−108
y=64±−92y=64±2i23y=32±i23
So, y=2+i233y = \{2 + i\sqrt{23}}{3}y=32+i23 and y=2−i233y = \{2 - i\sqrt{23}}
{3}y=32−i23.
10. Solving the equation:
(z−2)2−5(z+1)=0(z-2)^2 - 5(z+1) = 0(z−2)2−5(z+1)=0
Expand and simplify:
z2−4z+4−5z−5=0z2−9z−1=0z^2 - 4z + 4 - 5z - 5 = 0 \\ z^2 - 9z - 1 =
0z2−4z+4−5z−5=0z2−9z−1=0
Solve using the quadratic formula z=−b±b2−4ac2az = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}z=2a−b±b2−4ac:
z=9±81+42z=9±852z = \{9 \pm \sqrt{81 + 4}}{2} \\ z = \{9 \pm \sqrt{85}}{2}z=29±81+4
z=29±85
So, z=9+852z = \{9 + \sqrt{85}}{2}z=29+85 and z=9−852z = \{9 - \sqrt{85}}{2}z=29−85.
Advanced Algebraic Expressions
61. Simplify:
3x2−9x3x\frac{3x^2 - 9x}{3x}3x3x2−9x
Factor the numerator:
3x(x−3)3x=x−3\frac{3x(x - 3)}{3x} = x - 33x3x(x−3)=x−3
62. Simplify:
4y2−16y4y\frac{4y^2 - 16y}{4y}4y4y2−16y
Factor the numerator:
4y(y−4)4y=y−4\frac{4y(y - 4)}{4y} = y - 44y4y(y−4)=y−4
63. Simplify:
2z2+8z2z\frac{2z^2 + 8z}{2z}2z2z2+8z
Factor the numerator:
2z(z+4)2z=z+4\frac{2z(z + 4)}{2z} = z + 42z2z(z+4)=z+4
64. Expand:
(2x−3)2(2x - 3)^2(2x−3)2
Use the binomial expansion formula (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab +
b^2(a−b)2=a2−2ab+b2:
(2x−3)2=4x2−12x+9(2x - 3)^2 = 4x^2 - 12x + 9(2x−3)2=4x2−12x+9
65. Expand:
(y+4)2(y + 4)^2(y+4)2
Use the binomial expansion formula (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab +
b^2(a+b)2=a2+2ab+b2:
(y+4)2=y2+8y+16(y + 4)^2 = y^2 + 8y + 16(y+4)2=y2+8y+16
66. Expand:
(3a−2b)(3a+2b)(3a - 2b)(3a + 2b)(3a−2b)(3a+2b)
Use the difference of squares formula (a−b)(a+b)=a2−b2(a - b)(a + b) = a^2 - b^2(a−b)
(a+b)=a2−b2:
(3a−2b)(3a+2b)=9a2−4b2(3a - 2b)(3a + 2b) = 9a^2 - 4b^2(3a−2b)(3a+2b)=9a2−4b2
67. Factor:
x2−9x^2 - 9x2−9
This is a difference of squares:
5. Solving the equation:
3z−14+z+23=5\{3z-1}{4} + \{z+2}{3} = 543z−1+3z+2=5
Find a common denominator (12):
3(3z−1)12+4(z+2)12=59z−3+4z+812=513z+512=513z+5=6013z=55z=5513≈4.23\{3(3z-1)}
{12} + \{4(z+2)}{12} = 5 \\ \{9z - 3 + 4z + 8}{12} = 5 \\ \{13z + 5}{12} = 5 \\ 13z + 5 =
60 \\ 13z = 55 \\ z = \{55}{13} \approx 4.23123(3z−1)+124(z+2)=5129z−3+4z+8=51213z+5
=513z+5=6013z=55z=1355≈4.23
6. Solving the equation:
2(a−3)+3(a+4)=52(a-3) + 3(a+4) = 52(a−3)+3(a+4)=5
Expand and combine like terms:
2a−6+3a+12=55a+6=55a=−1a=−152a - 6 + 3a + 12 = 5 \\ 5a + 6 = 5 \\ 5a = -1 \\ a = -\{1}
{5}2a−6+3a+12=55a+6=55a=−1a=−51
7. Solving the equation:
5(b+2)−2(b−3)=105(b+2) - 2(b-3) = 105(b+2)−2(b−3)=10
Expand and combine like terms:
5b+10−2b+6=103b+16=103b=−6b=−25b + 10 - 2b + 6 = 10 \\ 3b + 16 = 10 \\ 3b = -6 \\ b = -
25b+10−2b+6=103b+16=103b=−6b=−2
8. Solving the equation:
(x+1)2−4(x−2)=0(x+1)^2 - 4(x-2) = 0(x+1)2−4(x−2)=0
Expand and simplify:
(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0(x+1)^2 - 4x + 8 = 0 \\ x^2 + 2x + 1 - 4x + 8 =
0 \\ x^2 - 2x + 9 = 0(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0
Solve using the quadratic formula x=−b±b2−4ac2ax = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}x=2a−b±b2−4ac:
x=2±4−362x=2±−322x=1±i8x = \{2 \pm \sqrt{4 - 36}}{2} \\ x = \{2 \pm \sqrt{-32}}{2} \\ x
= 1 \pm i\sqrt{8}x=22±4−36x=22±−32x=1±i8
So, x=1+22ix = 1 + 2\sqrt{2}ix=1+22i and x=1−22ix = 1 - 2\sqrt{2}ix=1−22i.
9. Solving the equation:
3(y−1)2+2(y+3)=03(y-1)^2 + 2(y+3) = 03(y−1)2+2(y+3)=0
Expand and combine like terms:
3(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=03(y^2 - 2y + 1) + 2y + 6 = 0 \\ 3y^2 - 6y
+ 3 + 2y + 6 = 0 \\ 3y^2 - 4y + 9 = 03(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=0
Solve using the quadratic formula y=−b±b2−4ac2ay = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}y=2a−b±b2−4ac:
y=4±16−1086y=4±−926y=4±2i236y=2±i233y = \{4 \pm \sqrt{16 - 108}}{6} \\ y = \{4 \pm \
sqrt{-92}}{6} \\ y = \{4 \pm 2i\sqrt{23}}{6} \\ y = \{2 \pm i\sqrt{23}}{3}y=64±16−108
y=64±−92y=64±2i23y=32±i23
So, y=2+i233y = \{2 + i\sqrt{23}}{3}y=32+i23 and y=2−i233y = \{2 - i\sqrt{23}}
{3}y=32−i23.
10. Solving the equation:
(z−2)2−5(z+1)=0(z-2)^2 - 5(z+1) = 0(z−2)2−5(z+1)=0
Expand and simplify:
z2−4z+4−5z−5=0z2−9z−1=0z^2 - 4z + 4 - 5z - 5 = 0 \\ z^2 - 9z - 1 =
0z2−4z+4−5z−5=0z2−9z−1=0
Solve using the quadratic formula z=−b±b2−4ac2az = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}z=2a−b±b2−4ac:
z=9±81+42z=9±852z = \{9 \pm \sqrt{81 + 4}}{2} \\ z = \{9 \pm \sqrt{85}}{2}z=29±81+4
z=29±85
So, z=9+852z = \{9 + \sqrt{85}}{2}z=29+85 and z=9−852z = \{9 - \sqrt{85}}{2}z=29−85.
Advanced Algebraic Expressions
61. Simplify:
3x2−9x3x\frac{3x^2 - 9x}{3x}3x3x2−9x
Factor the numerator:
3x(x−3)3x=x−3\frac{3x(x - 3)}{3x} = x - 33x3x(x−3)=x−3
62. Simplify:
4y2−16y4y\frac{4y^2 - 16y}{4y}4y4y2−16y
Factor the numerator:
4y(y−4)4y=y−4\frac{4y(y - 4)}{4y} = y - 44y4y(y−4)=y−4
63. Simplify:
2z2+8z2z\frac{2z^2 + 8z}{2z}2z2z2+8z
Factor the numerator:
2z(z+4)2z=z+4\frac{2z(z + 4)}{2z} = z + 42z2z(z+4)=z+4
64. Expand:
(2x−3)2(2x - 3)^2(2x−3)2
Use the binomial expansion formula (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab +
b^2(a−b)2=a2−2ab+b2:
(2x−3)2=4x2−12x+9(2x - 3)^2 = 4x^2 - 12x + 9(2x−3)2=4x2−12x+9
65. Expand:
(y+4)2(y + 4)^2(y+4)2
Use the binomial expansion formula (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab +
b^2(a+b)2=a2+2ab+b2:
(y+4)2=y2+8y+16(y + 4)^2 = y^2 + 8y + 16(y+4)2=y2+8y+16
66. Expand:
(3a−2b)(3a+2b)(3a - 2b)(3a + 2b)(3a−2b)(3a+2b)
Use the difference of squares formula (a−b)(a+b)=a2−b2(a - b)(a + b) = a^2 - b^2(a−b)
(a+b)=a2−b2:
(3a−2b)(3a+2b)=9a2−4b2(3a - 2b)(3a + 2b) = 9a^2 - 4b^2(3a−2b)(3a+2b)=9a2−4b2
67. Factor:
x2−9x^2 - 9x2−9
This is a difference of squares:
5. Solving the equation:
3z−14+z+23=5\{3z-1}{4} + \{z+2}{3} = 543z−1+3z+2=5
Find a common denominator (12):
3(3z−1)12+4(z+2)12=59z−3+4z+812=513z+512=513z+5=6013z=55z=5513≈4.23\{3(3z-1)}
{12} + \{4(z+2)}{12} = 5 \\ \{9z - 3 + 4z + 8}{12} = 5 \\ \{13z + 5}{12} = 5 \\ 13z + 5 =
60 \\ 13z = 55 \\ z = \{55}{13} \approx 4.23123(3z−1)+124(z+2)=5129z−3+4z+8=51213z+5
=513z+5=6013z=55z=1355≈4.23
6. Solving the equation:
2(a−3)+3(a+4)=52(a-3) + 3(a+4) = 52(a−3)+3(a+4)=5
Expand and combine like terms:
2a−6+3a+12=55a+6=55a=−1a=−152a - 6 + 3a + 12 = 5 \\ 5a + 6 = 5 \\ 5a = -1 \\ a = -\{1}
{5}2a−6+3a+12=55a+6=55a=−1a=−51
7. Solving the equation:
5(b+2)−2(b−3)=105(b+2) - 2(b-3) = 105(b+2)−2(b−3)=10
Expand and combine like terms:
5b+10−2b+6=103b+16=103b=−6b=−25b + 10 - 2b + 6 = 10 \\ 3b + 16 = 10 \\ 3b = -6 \\ b = -
25b+10−2b+6=103b+16=103b=−6b=−2
8. Solving the equation:
(x+1)2−4(x−2)=0(x+1)^2 - 4(x-2) = 0(x+1)2−4(x−2)=0
Expand and simplify:
(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0(x+1)^2 - 4x + 8 = 0 \\ x^2 + 2x + 1 - 4x + 8 =
0 \\ x^2 - 2x + 9 = 0(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0
Solve using the quadratic formula x=−b±b2−4ac2ax = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}x=2a−b±b2−4ac:
x=2±4−362x=2±−322x=1±i8x = \{2 \pm \sqrt{4 - 36}}{2} \\ x = \{2 \pm \sqrt{-32}}{2} \\ x
= 1 \pm i\sqrt{8}x=22±4−36x=22±−32x=1±i8
So, x=1+22ix = 1 + 2\sqrt{2}ix=1+22i and x=1−22ix = 1 - 2\sqrt{2}ix=1−22i.
9. Solving the equation:
3(y−1)2+2(y+3)=03(y-1)^2 + 2(y+3) = 03(y−1)2+2(y+3)=0
Expand and combine like terms:
3(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=03(y^2 - 2y + 1) + 2y + 6 = 0 \\ 3y^2 - 6y
+ 3 + 2y + 6 = 0 \\ 3y^2 - 4y + 9 = 03(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=0
Solve using the quadratic formula y=−b±b2−4ac2ay = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}y=2a−b±b2−4ac:
y=4±16−1086y=4±−926y=4±2i236y=2±i233y = \{4 \pm \sqrt{16 - 108}}{6} \\ y = \{4 \pm \
sqrt{-92}}{6} \\ y = \{4 \pm 2i\sqrt{23}}{6} \\ y = \{2 \pm i\sqrt{23}}{3}y=64±16−108
y=64±−92y=64±2i23y=32±i23
So, y=2+i233y = \{2 + i\sqrt{23}}{3}y=32+i23 and y=2−i233y = \{2 - i\sqrt{23}}
{3}y=32−i23.
10. Solving the equation:
(z−2)2−5(z+1)=0(z-2)^2 - 5(z+1) = 0(z−2)2−5(z+1)=0
Expand and simplify:
z2−4z+4−5z−5=0z2−9z−1=0z^2 - 4z + 4 - 5z - 5 = 0 \\ z^2 - 9z - 1 =
0z2−4z+4−5z−5=0z2−9z−1=0
Solve using the quadratic formula z=−b±b2−4ac2az = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}z=2a−b±b2−4ac:
z=9±81+42z=9±852z = \{9 \pm \sqrt{81 + 4}}{2} \\ z = \{9 \pm \sqrt{85}}{2}z=29±81+4
z=29±85
So, z=9+852z = \{9 + \sqrt{85}}{2}z=29+85 and z=9−852z = \{9 - \sqrt{85}}{2}z=29−85.
Advanced Algebraic Expressions
61. Simplify:
3x2−9x3x\frac{3x^2 - 9x}{3x}3x3x2−9x
Factor the numerator:
3x(x−3)3x=x−3\frac{3x(x - 3)}{3x} = x - 33x3x(x−3)=x−3
62. Simplify:
4y2−16y4y\frac{4y^2 - 16y}{4y}4y4y2−16y
Factor the numerator:
4y(y−4)4y=y−4\frac{4y(y - 4)}{4y} = y - 44y4y(y−4)=y−4
63. Simplify:
2z2+8z2z\frac{2z^2 + 8z}{2z}2z2z2+8z
Factor the numerator:
2z(z+4)2z=z+4\frac{2z(z + 4)}{2z} = z + 42z2z(z+4)=z+4
64. Expand:
(2x−3)2(2x - 3)^2(2x−3)2
Use the binomial expansion formula (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab +
b^2(a−b)2=a2−2ab+b2:
(2x−3)2=4x2−12x+9(2x - 3)^2 = 4x^2 - 12x + 9(2x−3)2=4x2−12x+9
65. Expand:
(y+4)2(y + 4)^2(y+4)2
Use the binomial expansion formula (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab +
b^2(a+b)2=a2+2ab+b2:
(y+4)2=y2+8y+16(y + 4)^2 = y^2 + 8y + 16(y+4)2=y2+8y+16
66. Expand:
(3a−2b)(3a+2b)(3a - 2b)(3a + 2b)(3a−2b)(3a+2b)
Use the difference of squares formula (a−b)(a+b)=a2−b2(a - b)(a + b) = a^2 - b^2(a−b)
(a+b)=a2−b2:
(3a−2b)(3a+2b)=9a2−4b2(3a - 2b)(3a + 2b) = 9a^2 - 4b^2(3a−2b)(3a+2b)=9a2−4b2
67. Factor:
x2−9x^2 - 9x2−9
This is a difference of squares:
5. Solving the equation:
3z−14+z+23=5\{3z-1}{4} + \{z+2}{3} = 543z−1+3z+2=5
Find a common denominator (12):
3(3z−1)12+4(z+2)12=59z−3+4z+812=513z+512=513z+5=6013z=55z=5513≈4.23\{3(3z-1)}
{12} + \{4(z+2)}{12} = 5 \\ \{9z - 3 + 4z + 8}{12} = 5 \\ \{13z + 5}{12} = 5 \\ 13z + 5 =
60 \\ 13z = 55 \\ z = \{55}{13} \approx 4.23123(3z−1)+124(z+2)=5129z−3+4z+8=51213z+5
=513z+5=6013z=55z=1355≈4.23
6. Solving the equation:
2(a−3)+3(a+4)=52(a-3) + 3(a+4) = 52(a−3)+3(a+4)=5
Expand and combine like terms:
2a−6+3a+12=55a+6=55a=−1a=−152a - 6 + 3a + 12 = 5 \\ 5a + 6 = 5 \\ 5a = -1 \\ a = -\{1}
{5}2a−6+3a+12=55a+6=55a=−1a=−51
7. Solving the equation:
5(b+2)−2(b−3)=105(b+2) - 2(b-3) = 105(b+2)−2(b−3)=10
Expand and combine like terms:
5b+10−2b+6=103b+16=103b=−6b=−25b + 10 - 2b + 6 = 10 \\ 3b + 16 = 10 \\ 3b = -6 \\ b = -
25b+10−2b+6=103b+16=103b=−6b=−2
8. Solving the equation:
(x+1)2−4(x−2)=0(x+1)^2 - 4(x-2) = 0(x+1)2−4(x−2)=0
Expand and simplify:
(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0(x+1)^2 - 4x + 8 = 0 \\ x^2 + 2x + 1 - 4x + 8 =
0 \\ x^2 - 2x + 9 = 0(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0
Solve using the quadratic formula x=−b±b2−4ac2ax = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}x=2a−b±b2−4ac:
x=2±4−362x=2±−322x=1±i8x = \{2 \pm \sqrt{4 - 36}}{2} \\ x = \{2 \pm \sqrt{-32}}{2} \\ x
= 1 \pm i\sqrt{8}x=22±4−36x=22±−32x=1±i8
So, x=1+22ix = 1 + 2\sqrt{2}ix=1+22i and x=1−22ix = 1 - 2\sqrt{2}ix=1−22i.
9. Solving the equation:
3(y−1)2+2(y+3)=03(y-1)^2 + 2(y+3) = 03(y−1)2+2(y+3)=0
Expand and combine like terms:
3(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=03(y^2 - 2y + 1) + 2y + 6 = 0 \\ 3y^2 - 6y
+ 3 + 2y + 6 = 0 \\ 3y^2 - 4y + 9 = 03(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=0
Solve using the quadratic formula y=−b±b2−4ac2ay = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}y=2a−b±b2−4ac:
y=4±16−1086y=4±−926y=4±2i236y=2±i233y = \{4 \pm \sqrt{16 - 108}}{6} \\ y = \{4 \pm \
sqrt{-92}}{6} \\ y = \{4 \pm 2i\sqrt{23}}{6} \\ y = \{2 \pm i\sqrt{23}}{3}y=64±16−108
y=64±−92y=64±2i23y=32±i23
So, y=2+i233y = \{2 + i\sqrt{23}}{3}y=32+i23 and y=2−i233y = \{2 - i\sqrt{23}}
{3}y=32−i23.
10. Solving the equation:
(z−2)2−5(z+1)=0(z-2)^2 - 5(z+1) = 0(z−2)2−5(z+1)=0
Expand and simplify:
z2−4z+4−5z−5=0z2−9z−1=0z^2 - 4z + 4 - 5z - 5 = 0 \\ z^2 - 9z - 1 =
0z2−4z+4−5z−5=0z2−9z−1=0
Solve using the quadratic formula z=−b±b2−4ac2az = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}z=2a−b±b2−4ac:
z=9±81+42z=9±852z = \{9 \pm \sqrt{81 + 4}}{2} \\ z = \{9 \pm \sqrt{85}}{2}z=29±81+4
z=29±85
So, z=9+852z = \{9 + \sqrt{85}}{2}z=29+85 and z=9−852z = \{9 - \sqrt{85}}{2}z=29−85.
Advanced Algebraic Expressions
61. Simplify:
3x2−9x3x\frac{3x^2 - 9x}{3x}3x3x2−9x
Factor the numerator:
3x(x−3)3x=x−3\frac{3x(x - 3)}{3x} = x - 33x3x(x−3)=x−3
62. Simplify:
4y2−16y4y\frac{4y^2 - 16y}{4y}4y4y2−16y
Factor the numerator:
4y(y−4)4y=y−4\frac{4y(y - 4)}{4y} = y - 44y4y(y−4)=y−4
63. Simplify:
2z2+8z2z\frac{2z^2 + 8z}{2z}2z2z2+8z
Factor the numerator:
2z(z+4)2z=z+4\frac{2z(z + 4)}{2z} = z + 42z2z(z+4)=z+4
64. Expand:
(2x−3)2(2x - 3)^2(2x−3)2
Use the binomial expansion formula (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab +
b^2(a−b)2=a2−2ab+b2:
(2x−3)2=4x2−12x+9(2x - 3)^2 = 4x^2 - 12x + 9(2x−3)2=4x2−12x+9
65. Expand:
(y+4)2(y + 4)^2(y+4)2
Use the binomial expansion formula (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab +
b^2(a+b)2=a2+2ab+b2:
(y+4)2=y2+8y+16(y + 4)^2 = y^2 + 8y + 16(y+4)2=y2+8y+16
66. Expand:
(3a−2b)(3a+2b)(3a - 2b)(3a + 2b)(3a−2b)(3a+2b)
Use the difference of squares formula (a−b)(a+b)=a2−b2(a - b)(a + b) = a^2 - b^2(a−b)
(a+b)=a2−b2:
(3a−2b)(3a+2b)=9a2−4b2(3a - 2b)(3a + 2b) = 9a^2 - 4b^2(3a−2b)(3a+2b)=9a2−4b2
67. Factor:
x2−9x^2 - 9x2−9
This is a difference of squares:
5. Solving the equation:
3z−14+z+23=5\{3z-1}{4} + \{z+2}{3} = 543z−1+3z+2=5
Find a common denominator (12):
3(3z−1)12+4(z+2)12=59z−3+4z+812=513z+512=513z+5=6013z=55z=5513≈4.23\{3(3z-1)}
{12} + \{4(z+2)}{12} = 5 \\ \{9z - 3 + 4z + 8}{12} = 5 \\ \{13z + 5}{12} = 5 \\ 13z + 5 =
60 \\ 13z = 55 \\ z = \{55}{13} \approx 4.23123(3z−1)+124(z+2)=5129z−3+4z+8=51213z+5
=513z+5=6013z=55z=1355≈4.23
6. Solving the equation:
2(a−3)+3(a+4)=52(a-3) + 3(a+4) = 52(a−3)+3(a+4)=5
Expand and combine like terms:
2a−6+3a+12=55a+6=55a=−1a=−152a - 6 + 3a + 12 = 5 \\ 5a + 6 = 5 \\ 5a = -1 \\ a = -\{1}
{5}2a−6+3a+12=55a+6=55a=−1a=−51
7. Solving the equation:
5(b+2)−2(b−3)=105(b+2) - 2(b-3) = 105(b+2)−2(b−3)=10
Expand and combine like terms:
5b+10−2b+6=103b+16=103b=−6b=−25b + 10 - 2b + 6 = 10 \\ 3b + 16 = 10 \\ 3b = -6 \\ b = -
25b+10−2b+6=103b+16=103b=−6b=−2
8. Solving the equation:
(x+1)2−4(x−2)=0(x+1)^2 - 4(x-2) = 0(x+1)2−4(x−2)=0
Expand and simplify:
(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0(x+1)^2 - 4x + 8 = 0 \\ x^2 + 2x + 1 - 4x + 8 =
0 \\ x^2 - 2x + 9 = 0(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0
Solve using the quadratic formula x=−b±b2−4ac2ax = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}x=2a−b±b2−4ac:
x=2±4−362x=2±−322x=1±i8x = \{2 \pm \sqrt{4 - 36}}{2} \\ x = \{2 \pm \sqrt{-32}}{2} \\ x
= 1 \pm i\sqrt{8}x=22±4−36x=22±−32x=1±i8
So, x=1+22ix = 1 + 2\sqrt{2}ix=1+22i and x=1−22ix = 1 - 2\sqrt{2}ix=1−22i.
9. Solving the equation:
3(y−1)2+2(y+3)=03(y-1)^2 + 2(y+3) = 03(y−1)2+2(y+3)=0
Expand and combine like terms:
3(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=03(y^2 - 2y + 1) + 2y + 6 = 0 \\ 3y^2 - 6y
+ 3 + 2y + 6 = 0 \\ 3y^2 - 4y + 9 = 03(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=0
Solve using the quadratic formula y=−b±b2−4ac2ay = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}y=2a−b±b2−4ac:
y=4±16−1086y=4±−926y=4±2i236y=2±i233y = \{4 \pm \sqrt{16 - 108}}{6} \\ y = \{4 \pm \
sqrt{-92}}{6} \\ y = \{4 \pm 2i\sqrt{23}}{6} \\ y = \{2 \pm i\sqrt{23}}{3}y=64±16−108
y=64±−92y=64±2i23y=32±i23
So, y=2+i233y = \{2 + i\sqrt{23}}{3}y=32+i23 and y=2−i233y = \{2 - i\sqrt{23}}
{3}y=32−i23.
10. Solving the equation:
(z−2)2−5(z+1)=0(z-2)^2 - 5(z+1) = 0(z−2)2−5(z+1)=0
Expand and simplify:
z2−4z+4−5z−5=0z2−9z−1=0z^2 - 4z + 4 - 5z - 5 = 0 \\ z^2 - 9z - 1 =
0z2−4z+4−5z−5=0z2−9z−1=0
Solve using the quadratic formula z=−b±b2−4ac2az = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}z=2a−b±b2−4ac:
z=9±81+42z=9±852z = \{9 \pm \sqrt{81 + 4}}{2} \\ z = \{9 \pm \sqrt{85}}{2}z=29±81+4
z=29±85
So, z=9+852z = \{9 + \sqrt{85}}{2}z=29+85 and z=9−852z = \{9 - \sqrt{85}}{2}z=29−85.
Advanced Algebraic Expressions
61. Simplify:
3x2−9x3x\frac{3x^2 - 9x}{3x}3x3x2−9x
Factor the numerator:
3x(x−3)3x=x−3\frac{3x(x - 3)}{3x} = x - 33x3x(x−3)=x−3
62. Simplify:
4y2−16y4y\frac{4y^2 - 16y}{4y}4y4y2−16y
Factor the numerator:
4y(y−4)4y=y−4\frac{4y(y - 4)}{4y} = y - 44y4y(y−4)=y−4
63. Simplify:
2z2+8z2z\frac{2z^2 + 8z}{2z}2z2z2+8z
Factor the numerator:
2z(z+4)2z=z+4\frac{2z(z + 4)}{2z} = z + 42z2z(z+4)=z+4
64. Expand:
(2x−3)2(2x - 3)^2(2x−3)2
Use the binomial expansion formula (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab +
b^2(a−b)2=a2−2ab+b2:
(2x−3)2=4x2−12x+9(2x - 3)^2 = 4x^2 - 12x + 9(2x−3)2=4x2−12x+9
65. Expand:
(y+4)2(y + 4)^2(y+4)2
Use the binomial expansion formula (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab +
b^2(a+b)2=a2+2ab+b2:
(y+4)2=y2+8y+16(y + 4)^2 = y^2 + 8y + 16(y+4)2=y2+8y+16
66. Expand:
(3a−2b)(3a+2b)(3a - 2b)(3a + 2b)(3a−2b)(3a+2b)
Use the difference of squares formula (a−b)(a+b)=a2−b2(a - b)(a + b) = a^2 - b^2(a−b)
(a+b)=a2−b2:
(3a−2b)(3a+2b)=9a2−4b2(3a - 2b)(3a + 2b) = 9a^2 - 4b^2(3a−2b)(3a+2b)=9a2−4b2
67. Factor:
x2−9x^2 - 9x2−9
This is a difference of squares:
5. Solving the equation:
3z−14+z+23=5\{3z-1}{4} + \{z+2}{3} = 543z−1+3z+2=5
Find a common denominator (12):
3(3z−1)12+4(z+2)12=59z−3+4z+812=513z+512=513z+5=6013z=55z=5513≈4.23\{3(3z-1)}
{12} + \{4(z+2)}{12} = 5 \\ \{9z - 3 + 4z + 8}{12} = 5 \\ \{13z + 5}{12} = 5 \\ 13z + 5 =
60 \\ 13z = 55 \\ z = \{55}{13} \approx 4.23123(3z−1)+124(z+2)=5129z−3+4z+8=51213z+5
=513z+5=6013z=55z=1355≈4.23
6. Solving the equation:
2(a−3)+3(a+4)=52(a-3) + 3(a+4) = 52(a−3)+3(a+4)=5
Expand and combine like terms:
2a−6+3a+12=55a+6=55a=−1a=−152a - 6 + 3a + 12 = 5 \\ 5a + 6 = 5 \\ 5a = -1 \\ a = -\{1}
{5}2a−6+3a+12=55a+6=55a=−1a=−51
7. Solving the equation:
5(b+2)−2(b−3)=105(b+2) - 2(b-3) = 105(b+2)−2(b−3)=10
Expand and combine like terms:
5b+10−2b+6=103b+16=103b=−6b=−25b + 10 - 2b + 6 = 10 \\ 3b + 16 = 10 \\ 3b = -6 \\ b = -
25b+10−2b+6=103b+16=103b=−6b=−2
8. Solving the equation:
(x+1)2−4(x−2)=0(x+1)^2 - 4(x-2) = 0(x+1)2−4(x−2)=0
Expand and simplify:
(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0(x+1)^2 - 4x + 8 = 0 \\ x^2 + 2x + 1 - 4x + 8 =
0 \\ x^2 - 2x + 9 = 0(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0
Solve using the quadratic formula x=−b±b2−4ac2ax = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}x=2a−b±b2−4ac:
x=2±4−362x=2±−322x=1±i8x = \{2 \pm \sqrt{4 - 36}}{2} \\ x = \{2 \pm \sqrt{-32}}{2} \\ x
= 1 \pm i\sqrt{8}x=22±4−36x=22±−32x=1±i8
So, x=1+22ix = 1 + 2\sqrt{2}ix=1+22i and x=1−22ix = 1 - 2\sqrt{2}ix=1−22i.
9. Solving the equation:
3(y−1)2+2(y+3)=03(y-1)^2 + 2(y+3) = 03(y−1)2+2(y+3)=0
Expand and combine like terms:
3(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=03(y^2 - 2y + 1) + 2y + 6 = 0 \\ 3y^2 - 6y
+ 3 + 2y + 6 = 0 \\ 3y^2 - 4y + 9 = 03(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=0
Solve using the quadratic formula y=−b±b2−4ac2ay = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}y=2a−b±b2−4ac:
y=4±16−1086y=4±−926y=4±2i236y=2±i233y = \{4 \pm \sqrt{16 - 108}}{6} \\ y = \{4 \pm \
sqrt{-92}}{6} \\ y = \{4 \pm 2i\sqrt{23}}{6} \\ y = \{2 \pm i\sqrt{23}}{3}y=64±16−108
y=64±−92y=64±2i23y=32±i23
So, y=2+i233y = \{2 + i\sqrt{23}}{3}y=32+i23 and y=2−i233y = \{2 - i\sqrt{23}}
{3}y=32−i23.
10. Solving the equation:
(z−2)2−5(z+1)=0(z-2)^2 - 5(z+1) = 0(z−2)2−5(z+1)=0
Expand and simplify:
z2−4z+4−5z−5=0z2−9z−1=0z^2 - 4z + 4 - 5z - 5 = 0 \\ z^2 - 9z - 1 =
0z2−4z+4−5z−5=0z2−9z−1=0
Solve using the quadratic formula z=−b±b2−4ac2az = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}z=2a−b±b2−4ac:
z=9±81+42z=9±852z = \{9 \pm \sqrt{81 + 4}}{2} \\ z = \{9 \pm \sqrt{85}}{2}z=29±81+4
z=29±85
So, z=9+852z = \{9 + \sqrt{85}}{2}z=29+85 and z=9−852z = \{9 - \sqrt{85}}{2}z=29−85.
Advanced Algebraic Expressions
61. Simplify:
3x2−9x3x\frac{3x^2 - 9x}{3x}3x3x2−9x
Factor the numerator:
3x(x−3)3x=x−3\frac{3x(x - 3)}{3x} = x - 33x3x(x−3)=x−3
62. Simplify:
4y2−16y4y\frac{4y^2 - 16y}{4y}4y4y2−16y
Factor the numerator:
4y(y−4)4y=y−4\frac{4y(y - 4)}{4y} = y - 44y4y(y−4)=y−4
63. Simplify:
2z2+8z2z\frac{2z^2 + 8z}{2z}2z2z2+8z
Factor the numerator:
2z(z+4)2z=z+4\frac{2z(z + 4)}{2z} = z + 42z2z(z+4)=z+4
64. Expand:
(2x−3)2(2x - 3)^2(2x−3)2
Use the binomial expansion formula (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab +
b^2(a−b)2=a2−2ab+b2:
(2x−3)2=4x2−12x+9(2x - 3)^2 = 4x^2 - 12x + 9(2x−3)2=4x2−12x+9
65. Expand:
(y+4)2(y + 4)^2(y+4)2
Use the binomial expansion formula (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab +
b^2(a+b)2=a2+2ab+b2:
(y+4)2=y2+8y+16(y + 4)^2 = y^2 + 8y + 16(y+4)2=y2+8y+16
66. Expand:
(3a−2b)(3a+2b)(3a - 2b)(3a + 2b)(3a−2b)(3a+2b)
Use the difference of squares formula (a−b)(a+b)=a2−b2(a - b)(a + b) = a^2 - b^2(a−b)
(a+b)=a2−b2:
(3a−2b)(3a+2b)=9a2−4b2(3a - 2b)(3a + 2b) = 9a^2 - 4b^2(3a−2b)(3a+2b)=9a2−4b2
67. Factor:
x2−9x^2 - 9x2−9
This is a difference of squares:
5. Solving the equation:
3z−14+z+23=5\{3z-1}{4} + \{z+2}{3} = 543z−1+3z+2=5
Find a common denominator (12):
3(3z−1)12+4(z+2)12=59z−3+4z+812=513z+512=513z+5=6013z=55z=5513≈4.23\{3(3z-1)}
{12} + \{4(z+2)}{12} = 5 \\ \{9z - 3 + 4z + 8}{12} = 5 \\ \{13z + 5}{12} = 5 \\ 13z + 5 =
60 \\ 13z = 55 \\ z = \{55}{13} \approx 4.23123(3z−1)+124(z+2)=5129z−3+4z+8=51213z+5
=513z+5=6013z=55z=1355≈4.23
6. Solving the equation:
2(a−3)+3(a+4)=52(a-3) + 3(a+4) = 52(a−3)+3(a+4)=5
Expand and combine like terms:
2a−6+3a+12=55a+6=55a=−1a=−152a - 6 + 3a + 12 = 5 \\ 5a + 6 = 5 \\ 5a = -1 \\ a = -\{1}
{5}2a−6+3a+12=55a+6=55a=−1a=−51
7. Solving the equation:
5(b+2)−2(b−3)=105(b+2) - 2(b-3) = 105(b+2)−2(b−3)=10
Expand and combine like terms:
5b+10−2b+6=103b+16=103b=−6b=−25b + 10 - 2b + 6 = 10 \\ 3b + 16 = 10 \\ 3b = -6 \\ b = -
25b+10−2b+6=103b+16=103b=−6b=−2
8. Solving the equation:
(x+1)2−4(x−2)=0(x+1)^2 - 4(x-2) = 0(x+1)2−4(x−2)=0
Expand and simplify:
(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0(x+1)^2 - 4x + 8 = 0 \\ x^2 + 2x + 1 - 4x + 8 =
0 \\ x^2 - 2x + 9 = 0(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0
Solve using the quadratic formula x=−b±b2−4ac2ax = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}x=2a−b±b2−4ac:
x=2±4−362x=2±−322x=1±i8x = \{2 \pm \sqrt{4 - 36}}{2} \\ x = \{2 \pm \sqrt{-32}}{2} \\ x
= 1 \pm i\sqrt{8}x=22±4−36x=22±−32x=1±i8
So, x=1+22ix = 1 + 2\sqrt{2}ix=1+22i and x=1−22ix = 1 - 2\sqrt{2}ix=1−22i.
9. Solving the equation:
3(y−1)2+2(y+3)=03(y-1)^2 + 2(y+3) = 03(y−1)2+2(y+3)=0
Expand and combine like terms:
3(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=03(y^2 - 2y + 1) + 2y + 6 = 0 \\ 3y^2 - 6y
+ 3 + 2y + 6 = 0 \\ 3y^2 - 4y + 9 = 03(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=0
Solve using the quadratic formula y=−b±b2−4ac2ay = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}y=2a−b±b2−4ac:
y=4±16−1086y=4±−926y=4±2i236y=2±i233y = \{4 \pm \sqrt{16 - 108}}{6} \\ y = \{4 \pm \
sqrt{-92}}{6} \\ y = \{4 \pm 2i\sqrt{23}}{6} \\ y = \{2 \pm i\sqrt{23}}{3}y=64±16−108
y=64±−92y=64±2i23y=32±i23
So, y=2+i233y = \{2 + i\sqrt{23}}{3}y=32+i23 and y=2−i233y = \{2 - i\sqrt{23}}
{3}y=32−i23.
10. Solving the equation:
(z−2)2−5(z+1)=0(z-2)^2 - 5(z+1) = 0(z−2)2−5(z+1)=0
Expand and simplify:
z2−4z+4−5z−5=0z2−9z−1=0z^2 - 4z + 4 - 5z - 5 = 0 \\ z^2 - 9z - 1 =
0z2−4z+4−5z−5=0z2−9z−1=0
Solve using the quadratic formula z=−b±b2−4ac2az = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}z=2a−b±b2−4ac:
z=9±81+42z=9±852z = \{9 \pm \sqrt{81 + 4}}{2} \\ z = \{9 \pm \sqrt{85}}{2}z=29±81+4
z=29±85
So, z=9+852z = \{9 + \sqrt{85}}{2}z=29+85 and z=9−852z = \{9 - \sqrt{85}}{2}z=29−85.
Advanced Algebraic Expressions
61. Simplify:
3x2−9x3x\frac{3x^2 - 9x}{3x}3x3x2−9x
Factor the numerator:
3x(x−3)3x=x−3\frac{3x(x - 3)}{3x} = x - 33x3x(x−3)=x−3
62. Simplify:
4y2−16y4y\frac{4y^2 - 16y}{4y}4y4y2−16y
Factor the numerator:
4y(y−4)4y=y−4\frac{4y(y - 4)}{4y} = y - 44y4y(y−4)=y−4
63. Simplify:
2z2+8z2z\frac{2z^2 + 8z}{2z}2z2z2+8z
Factor the numerator:
2z(z+4)2z=z+4\frac{2z(z + 4)}{2z} = z + 42z2z(z+4)=z+4
64. Expand:
(2x−3)2(2x - 3)^2(2x−3)2
Use the binomial expansion formula (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab +
b^2(a−b)2=a2−2ab+b2:
(2x−3)2=4x2−12x+9(2x - 3)^2 = 4x^2 - 12x + 9(2x−3)2=4x2−12x+9
65. Expand:
(y+4)2(y + 4)^2(y+4)2
Use the binomial expansion formula (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab +
b^2(a+b)2=a2+2ab+b2:
(y+4)2=y2+8y+16(y + 4)^2 = y^2 + 8y + 16(y+4)2=y2+8y+16
66. Expand:
(3a−2b)(3a+2b)(3a - 2b)(3a + 2b)(3a−2b)(3a+2b)
Use the difference of squares formula (a−b)(a+b)=a2−b2(a - b)(a + b) = a^2 - b^2(a−b)
(a+b)=a2−b2:
(3a−2b)(3a+2b)=9a2−4b2(3a - 2b)(3a + 2b) = 9a^2 - 4b^2(3a−2b)(3a+2b)=9a2−4b2
67. Factor:
x2−9x^2 - 9x2−9
This is a difference of squares:
5. Solving the equation:
3z−14+z+23=5\{3z-1}{4} + \{z+2}{3} = 543z−1+3z+2=5
Find a common denominator (12):
3(3z−1)12+4(z+2)12=59z−3+4z+812=513z+512=513z+5=6013z=55z=5513≈4.23\{3(3z-1)}
{12} + \{4(z+2)}{12} = 5 \\ \{9z - 3 + 4z + 8}{12} = 5 \\ \{13z + 5}{12} = 5 \\ 13z + 5 =
60 \\ 13z = 55 \\ z = \{55}{13} \approx 4.23123(3z−1)+124(z+2)=5129z−3+4z+8=51213z+5
=513z+5=6013z=55z=1355≈4.23
6. Solving the equation:
2(a−3)+3(a+4)=52(a-3) + 3(a+4) = 52(a−3)+3(a+4)=5
Expand and combine like terms:
2a−6+3a+12=55a+6=55a=−1a=−152a - 6 + 3a + 12 = 5 \\ 5a + 6 = 5 \\ 5a = -1 \\ a = -\{1}
{5}2a−6+3a+12=55a+6=55a=−1a=−51
7. Solving the equation:
5(b+2)−2(b−3)=105(b+2) - 2(b-3) = 105(b+2)−2(b−3)=10
Expand and combine like terms:
5b+10−2b+6=103b+16=103b=−6b=−25b + 10 - 2b + 6 = 10 \\ 3b + 16 = 10 \\ 3b = -6 \\ b = -
25b+10−2b+6=103b+16=103b=−6b=−2
8. Solving the equation:
(x+1)2−4(x−2)=0(x+1)^2 - 4(x-2) = 0(x+1)2−4(x−2)=0
Expand and simplify:
(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0(x+1)^2 - 4x + 8 = 0 \\ x^2 + 2x + 1 - 4x + 8 =
0 \\ x^2 - 2x + 9 = 0(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0
Solve using the quadratic formula x=−b±b2−4ac2ax = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}x=2a−b±b2−4ac:
x=2±4−362x=2±−322x=1±i8x = \{2 \pm \sqrt{4 - 36}}{2} \\ x = \{2 \pm \sqrt{-32}}{2} \\ x
= 1 \pm i\sqrt{8}x=22±4−36x=22±−32x=1±i8
So, x=1+22ix = 1 + 2\sqrt{2}ix=1+22i and x=1−22ix = 1 - 2\sqrt{2}ix=1−22i.
9. Solving the equation:
3(y−1)2+2(y+3)=03(y-1)^2 + 2(y+3) = 03(y−1)2+2(y+3)=0
Expand and combine like terms:
3(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=03(y^2 - 2y + 1) + 2y + 6 = 0 \\ 3y^2 - 6y
+ 3 + 2y + 6 = 0 \\ 3y^2 - 4y + 9 = 03(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=0
Solve using the quadratic formula y=−b±b2−4ac2ay = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}y=2a−b±b2−4ac:
y=4±16−1086y=4±−926y=4±2i236y=2±i233y = \{4 \pm \sqrt{16 - 108}}{6} \\ y = \{4 \pm \
sqrt{-92}}{6} \\ y = \{4 \pm 2i\sqrt{23}}{6} \\ y = \{2 \pm i\sqrt{23}}{3}y=64±16−108
y=64±−92y=64±2i23y=32±i23
So, y=2+i233y = \{2 + i\sqrt{23}}{3}y=32+i23 and y=2−i233y = \{2 - i\sqrt{23}}
{3}y=32−i23.
10. Solving the equation:
(z−2)2−5(z+1)=0(z-2)^2 - 5(z+1) = 0(z−2)2−5(z+1)=0
Expand and simplify:
z2−4z+4−5z−5=0z2−9z−1=0z^2 - 4z + 4 - 5z - 5 = 0 \\ z^2 - 9z - 1 =
0z2−4z+4−5z−5=0z2−9z−1=0
Solve using the quadratic formula z=−b±b2−4ac2az = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}z=2a−b±b2−4ac:
z=9±81+42z=9±852z = \{9 \pm \sqrt{81 + 4}}{2} \\ z = \{9 \pm \sqrt{85}}{2}z=29±81+4
z=29±85
So, z=9+852z = \{9 + \sqrt{85}}{2}z=29+85 and z=9−852z = \{9 - \sqrt{85}}{2}z=29−85.
Advanced Algebraic Expressions
61. Simplify:
3x2−9x3x\frac{3x^2 - 9x}{3x}3x3x2−9x
Factor the numerator:
3x(x−3)3x=x−3\frac{3x(x - 3)}{3x} = x - 33x3x(x−3)=x−3
62. Simplify:
4y2−16y4y\frac{4y^2 - 16y}{4y}4y4y2−16y
Factor the numerator:
4y(y−4)4y=y−4\frac{4y(y - 4)}{4y} = y - 44y4y(y−4)=y−4
63. Simplify:
2z2+8z2z\frac{2z^2 + 8z}{2z}2z2z2+8z
Factor the numerator:
2z(z+4)2z=z+4\frac{2z(z + 4)}{2z} = z + 42z2z(z+4)=z+4
64. Expand:
(2x−3)2(2x - 3)^2(2x−3)2
Use the binomial expansion formula (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab +
b^2(a−b)2=a2−2ab+b2:
(2x−3)2=4x2−12x+9(2x - 3)^2 = 4x^2 - 12x + 9(2x−3)2=4x2−12x+9
65. Expand:
(y+4)2(y + 4)^2(y+4)2
Use the binomial expansion formula (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab +
b^2(a+b)2=a2+2ab+b2:
(y+4)2=y2+8y+16(y + 4)^2 = y^2 + 8y + 16(y+4)2=y2+8y+16
66. Expand:
(3a−2b)(3a+2b)(3a - 2b)(3a + 2b)(3a−2b)(3a+2b)
Use the difference of squares formula (a−b)(a+b)=a2−b2(a - b)(a + b) = a^2 - b^2(a−b)
(a+b)=a2−b2:
(3a−2b)(3a+2b)=9a2−4b2(3a - 2b)(3a + 2b) = 9a^2 - 4b^2(3a−2b)(3a+2b)=9a2−4b2
67. Factor:
x2−9x^2 - 9x2−9
This is a difference of squares:
5. Solving the equation:
3z−14+z+23=5\{3z-1}{4} + \{z+2}{3} = 543z−1+3z+2=5
Find a common denominator (12):
3(3z−1)12+4(z+2)12=59z−3+4z+812=513z+512=513z+5=6013z=55z=5513≈4.23\{3(3z-1)}
{12} + \{4(z+2)}{12} = 5 \\ \{9z - 3 + 4z + 8}{12} = 5 \\ \{13z + 5}{12} = 5 \\ 13z + 5 =
60 \\ 13z = 55 \\ z = \{55}{13} \approx 4.23123(3z−1)+124(z+2)=5129z−3+4z+8=51213z+5
=513z+5=6013z=55z=1355≈4.23
6. Solving the equation:
2(a−3)+3(a+4)=52(a-3) + 3(a+4) = 52(a−3)+3(a+4)=5
Expand and combine like terms:
2a−6+3a+12=55a+6=55a=−1a=−152a - 6 + 3a + 12 = 5 \\ 5a + 6 = 5 \\ 5a = -1 \\ a = -\{1}
{5}2a−6+3a+12=55a+6=55a=−1a=−51
7. Solving the equation:
5(b+2)−2(b−3)=105(b+2) - 2(b-3) = 105(b+2)−2(b−3)=10
Expand and combine like terms:
5b+10−2b+6=103b+16=103b=−6b=−25b + 10 - 2b + 6 = 10 \\ 3b + 16 = 10 \\ 3b = -6 \\ b = -
25b+10−2b+6=103b+16=103b=−6b=−2
8. Solving the equation:
(x+1)2−4(x−2)=0(x+1)^2 - 4(x-2) = 0(x+1)2−4(x−2)=0
Expand and simplify:
(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0(x+1)^2 - 4x + 8 = 0 \\ x^2 + 2x + 1 - 4x + 8 =
0 \\ x^2 - 2x + 9 = 0(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0
Solve using the quadratic formula x=−b±b2−4ac2ax = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}x=2a−b±b2−4ac:
x=2±4−362x=2±−322x=1±i8x = \{2 \pm \sqrt{4 - 36}}{2} \\ x = \{2 \pm \sqrt{-32}}{2} \\ x
= 1 \pm i\sqrt{8}x=22±4−36x=22±−32x=1±i8
So, x=1+22ix = 1 + 2\sqrt{2}ix=1+22i and x=1−22ix = 1 - 2\sqrt{2}ix=1−22i.
9. Solving the equation:
3(y−1)2+2(y+3)=03(y-1)^2 + 2(y+3) = 03(y−1)2+2(y+3)=0
Expand and combine like terms:
3(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=03(y^2 - 2y + 1) + 2y + 6 = 0 \\ 3y^2 - 6y
+ 3 + 2y + 6 = 0 \\ 3y^2 - 4y + 9 = 03(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=0
Solve using the quadratic formula y=−b±b2−4ac2ay = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}y=2a−b±b2−4ac:
y=4±16−1086y=4±−926y=4±2i236y=2±i233y = \{4 \pm \sqrt{16 - 108}}{6} \\ y = \{4 \pm \
sqrt{-92}}{6} \\ y = \{4 \pm 2i\sqrt{23}}{6} \\ y = \{2 \pm i\sqrt{23}}{3}y=64±16−108
y=64±−92y=64±2i23y=32±i23
So, y=2+i233y = \{2 + i\sqrt{23}}{3}y=32+i23 and y=2−i233y = \{2 - i\sqrt{23}}
{3}y=32−i23.
10. Solving the equation:
(z−2)2−5(z+1)=0(z-2)^2 - 5(z+1) = 0(z−2)2−5(z+1)=0
Expand and simplify:
z2−4z+4−5z−5=0z2−9z−1=0z^2 - 4z + 4 - 5z - 5 = 0 \\ z^2 - 9z - 1 =
0z2−4z+4−5z−5=0z2−9z−1=0
Solve using the quadratic formula z=−b±b2−4ac2az = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}z=2a−b±b2−4ac:
z=9±81+42z=9±852z = \{9 \pm \sqrt{81 + 4}}{2} \\ z = \{9 \pm \sqrt{85}}{2}z=29±81+4
z=29±85
So, z=9+852z = \{9 + \sqrt{85}}{2}z=29+85 and z=9−852z = \{9 - \sqrt{85}}{2}z=29−85.
Advanced Algebraic Expressions
61. Simplify:
3x2−9x3x\frac{3x^2 - 9x}{3x}3x3x2−9x
Factor the numerator:
3x(x−3)3x=x−3\frac{3x(x - 3)}{3x} = x - 33x3x(x−3)=x−3
62. Simplify:
4y2−16y4y\frac{4y^2 - 16y}{4y}4y4y2−16y
Factor the numerator:
4y(y−4)4y=y−4\frac{4y(y - 4)}{4y} = y - 44y4y(y−4)=y−4
63. Simplify:
2z2+8z2z\frac{2z^2 + 8z}{2z}2z2z2+8z
Factor the numerator:
2z(z+4)2z=z+4\frac{2z(z + 4)}{2z} = z + 42z2z(z+4)=z+4
64. Expand:
(2x−3)2(2x - 3)^2(2x−3)2
Use the binomial expansion formula (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab +
b^2(a−b)2=a2−2ab+b2:
(2x−3)2=4x2−12x+9(2x - 3)^2 = 4x^2 - 12x + 9(2x−3)2=4x2−12x+9
65. Expand:
(y+4)2(y + 4)^2(y+4)2
Use the binomial expansion formula (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab +
b^2(a+b)2=a2+2ab+b2:
(y+4)2=y2+8y+16(y + 4)^2 = y^2 + 8y + 16(y+4)2=y2+8y+16
66. Expand:
(3a−2b)(3a+2b)(3a - 2b)(3a + 2b)(3a−2b)(3a+2b)
Use the difference of squares formula (a−b)(a+b)=a2−b2(a - b)(a + b) = a^2 - b^2(a−b)
(a+b)=a2−b2:
(3a−2b)(3a+2b)=9a2−4b2(3a - 2b)(3a + 2b) = 9a^2 - 4b^2(3a−2b)(3a+2b)=9a2−4b2
67. Factor:
x2−9x^2 - 9x2−9
This is a difference of squares:
5. Solving the equation:
3z−14+z+23=5\{3z-1}{4} + \{z+2}{3} = 543z−1+3z+2=5
Find a common denominator (12):
3(3z−1)12+4(z+2)12=59z−3+4z+812=513z+512=513z+5=6013z=55z=5513≈4.23\{3(3z-1)}
{12} + \{4(z+2)}{12} = 5 \\ \{9z - 3 + 4z + 8}{12} = 5 \\ \{13z + 5}{12} = 5 \\ 13z + 5 =
60 \\ 13z = 55 \\ z = \{55}{13} \approx 4.23123(3z−1)+124(z+2)=5129z−3+4z+8=51213z+5
=513z+5=6013z=55z=1355≈4.23
6. Solving the equation:
2(a−3)+3(a+4)=52(a-3) + 3(a+4) = 52(a−3)+3(a+4)=5
Expand and combine like terms:
2a−6+3a+12=55a+6=55a=−1a=−152a - 6 + 3a + 12 = 5 \\ 5a + 6 = 5 \\ 5a = -1 \\ a = -\{1}
{5}2a−6+3a+12=55a+6=55a=−1a=−51
7. Solving the equation:
5(b+2)−2(b−3)=105(b+2) - 2(b-3) = 105(b+2)−2(b−3)=10
Expand and combine like terms:
5b+10−2b+6=103b+16=103b=−6b=−25b + 10 - 2b + 6 = 10 \\ 3b + 16 = 10 \\ 3b = -6 \\ b = -
25b+10−2b+6=103b+16=103b=−6b=−2
8. Solving the equation:
(x+1)2−4(x−2)=0(x+1)^2 - 4(x-2) = 0(x+1)2−4(x−2)=0
Expand and simplify:
(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0(x+1)^2 - 4x + 8 = 0 \\ x^2 + 2x + 1 - 4x + 8 =
0 \\ x^2 - 2x + 9 = 0(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0
Solve using the quadratic formula x=−b±b2−4ac2ax = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}x=2a−b±b2−4ac:
x=2±4−362x=2±−322x=1±i8x = \{2 \pm \sqrt{4 - 36}}{2} \\ x = \{2 \pm \sqrt{-32}}{2} \\ x
= 1 \pm i\sqrt{8}x=22±4−36x=22±−32x=1±i8
So, x=1+22ix = 1 + 2\sqrt{2}ix=1+22i and x=1−22ix = 1 - 2\sqrt{2}ix=1−22i.
9. Solving the equation:
3(y−1)2+2(y+3)=03(y-1)^2 + 2(y+3) = 03(y−1)2+2(y+3)=0
Expand and combine like terms:
3(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=03(y^2 - 2y + 1) + 2y + 6 = 0 \\ 3y^2 - 6y
+ 3 + 2y + 6 = 0 \\ 3y^2 - 4y + 9 = 03(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=0
Solve using the quadratic formula y=−b±b2−4ac2ay = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}y=2a−b±b2−4ac:
y=4±16−1086y=4±−926y=4±2i236y=2±i233y = \{4 \pm \sqrt{16 - 108}}{6} \\ y = \{4 \pm \
sqrt{-92}}{6} \\ y = \{4 \pm 2i\sqrt{23}}{6} \\ y = \{2 \pm i\sqrt{23}}{3}y=64±16−108
y=64±−92y=64±2i23y=32±i23
So, y=2+i233y = \{2 + i\sqrt{23}}{3}y=32+i23 and y=2−i233y = \{2 - i\sqrt{23}}
{3}y=32−i23.
10. Solving the equation:
(z−2)2−5(z+1)=0(z-2)^2 - 5(z+1) = 0(z−2)2−5(z+1)=0
Expand and simplify:
z2−4z+4−5z−5=0z2−9z−1=0z^2 - 4z + 4 - 5z - 5 = 0 \\ z^2 - 9z - 1 =
0z2−4z+4−5z−5=0z2−9z−1=0
Solve using the quadratic formula z=−b±b2−4ac2az = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}z=2a−b±b2−4ac:
z=9±81+42z=9±852z = \{9 \pm \sqrt{81 + 4}}{2} \\ z = \{9 \pm \sqrt{85}}{2}z=29±81+4
z=29±85
So, z=9+852z = \{9 + \sqrt{85}}{2}z=29+85 and z=9−852z = \{9 - \sqrt{85}}{2}z=29−85.
Advanced Algebraic Expressions
61. Simplify:
3x2−9x3x\frac{3x^2 - 9x}{3x}3x3x2−9x
Factor the numerator:
3x(x−3)3x=x−3\frac{3x(x - 3)}{3x} = x - 33x3x(x−3)=x−3
62. Simplify:
4y2−16y4y\frac{4y^2 - 16y}{4y}4y4y2−16y
Factor the numerator:
4y(y−4)4y=y−4\frac{4y(y - 4)}{4y} = y - 44y4y(y−4)=y−4
63. Simplify:
2z2+8z2z\frac{2z^2 + 8z}{2z}2z2z2+8z
Factor the numerator:
2z(z+4)2z=z+4\frac{2z(z + 4)}{2z} = z + 42z2z(z+4)=z+4
64. Expand:
(2x−3)2(2x - 3)^2(2x−3)2
Use the binomial expansion formula (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab +
b^2(a−b)2=a2−2ab+b2:
(2x−3)2=4x2−12x+9(2x - 3)^2 = 4x^2 - 12x + 9(2x−3)2=4x2−12x+9
65. Expand:
(y+4)2(y + 4)^2(y+4)2
Use the binomial expansion formula (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab +
b^2(a+b)2=a2+2ab+b2:
(y+4)2=y2+8y+16(y + 4)^2 = y^2 + 8y + 16(y+4)2=y2+8y+16
66. Expand:
(3a−2b)(3a+2b)(3a - 2b)(3a + 2b)(3a−2b)(3a+2b)
Use the difference of squares formula (a−b)(a+b)=a2−b2(a - b)(a + b) = a^2 - b^2(a−b)
(a+b)=a2−b2:
(3a−2b)(3a+2b)=9a2−4b2(3a - 2b)(3a + 2b) = 9a^2 - 4b^2(3a−2b)(3a+2b)=9a2−4b2
67. Factor:
x2−9x^2 - 9x2−9
This is a difference of squares:
5. Solving the equation:
3z−14+z+23=5\{3z-1}{4} + \{z+2}{3} = 543z−1+3z+2=5
Find a common denominator (12):
3(3z−1)12+4(z+2)12=59z−3+4z+812=513z+512=513z+5=6013z=55z=5513≈4.23\{3(3z-1)}
{12} + \{4(z+2)}{12} = 5 \\ \{9z - 3 + 4z + 8}{12} = 5 \\ \{13z + 5}{12} = 5 \\ 13z + 5 =
60 \\ 13z = 55 \\ z = \{55}{13} \approx 4.23123(3z−1)+124(z+2)=5129z−3+4z+8=51213z+5
=513z+5=6013z=55z=1355≈4.23
6. Solving the equation:
2(a−3)+3(a+4)=52(a-3) + 3(a+4) = 52(a−3)+3(a+4)=5
Expand and combine like terms:
2a−6+3a+12=55a+6=55a=−1a=−152a - 6 + 3a + 12 = 5 \\ 5a + 6 = 5 \\ 5a = -1 \\ a = -\{1}
{5}2a−6+3a+12=55a+6=55a=−1a=−51
7. Solving the equation:
5(b+2)−2(b−3)=105(b+2) - 2(b-3) = 105(b+2)−2(b−3)=10
Expand and combine like terms:
5b+10−2b+6=103b+16=103b=−6b=−25b + 10 - 2b + 6 = 10 \\ 3b + 16 = 10 \\ 3b = -6 \\ b = -
25b+10−2b+6=103b+16=103b=−6b=−2
8. Solving the equation:
(x+1)2−4(x−2)=0(x+1)^2 - 4(x-2) = 0(x+1)2−4(x−2)=0
Expand and simplify:
(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0(x+1)^2 - 4x + 8 = 0 \\ x^2 + 2x + 1 - 4x + 8 =
0 \\ x^2 - 2x + 9 = 0(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0
Solve using the quadratic formula x=−b±b2−4ac2ax = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}x=2a−b±b2−4ac:
x=2±4−362x=2±−322x=1±i8x = \{2 \pm \sqrt{4 - 36}}{2} \\ x = \{2 \pm \sqrt{-32}}{2} \\ x
= 1 \pm i\sqrt{8}x=22±4−36x=22±−32x=1±i8
So, x=1+22ix = 1 + 2\sqrt{2}ix=1+22i and x=1−22ix = 1 - 2\sqrt{2}ix=1−22i.
9. Solving the equation:
3(y−1)2+2(y+3)=03(y-1)^2 + 2(y+3) = 03(y−1)2+2(y+3)=0
Expand and combine like terms:
3(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=03(y^2 - 2y + 1) + 2y + 6 = 0 \\ 3y^2 - 6y
+ 3 + 2y + 6 = 0 \\ 3y^2 - 4y + 9 = 03(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=0
Solve using the quadratic formula y=−b±b2−4ac2ay = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}y=2a−b±b2−4ac:
y=4±16−1086y=4±−926y=4±2i236y=2±i233y = \{4 \pm \sqrt{16 - 108}}{6} \\ y = \{4 \pm \
sqrt{-92}}{6} \\ y = \{4 \pm 2i\sqrt{23}}{6} \\ y = \{2 \pm i\sqrt{23}}{3}y=64±16−108
y=64±−92y=64±2i23y=32±i23
So, y=2+i233y = \{2 + i\sqrt{23}}{3}y=32+i23 and y=2−i233y = \{2 - i\sqrt{23}}
{3}y=32−i23.
10. Solving the equation:
(z−2)2−5(z+1)=0(z-2)^2 - 5(z+1) = 0(z−2)2−5(z+1)=0
Expand and simplify:
z2−4z+4−5z−5=0z2−9z−1=0z^2 - 4z + 4 - 5z - 5 = 0 \\ z^2 - 9z - 1 =
0z2−4z+4−5z−5=0z2−9z−1=0
Solve using the quadratic formula z=−b±b2−4ac2az = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}z=2a−b±b2−4ac:
z=9±81+42z=9±852z = \{9 \pm \sqrt{81 + 4}}{2} \\ z = \{9 \pm \sqrt{85}}{2}z=29±81+4
z=29±85
So, z=9+852z = \{9 + \sqrt{85}}{2}z=29+85 and z=9−852z = \{9 - \sqrt{85}}{2}z=29−85.
Advanced Algebraic Expressions
61. Simplify:
3x2−9x3x\frac{3x^2 - 9x}{3x}3x3x2−9x
Factor the numerator:
3x(x−3)3x=x−3\frac{3x(x - 3)}{3x} = x - 33x3x(x−3)=x−3
62. Simplify:
4y2−16y4y\frac{4y^2 - 16y}{4y}4y4y2−16y
Factor the numerator:
4y(y−4)4y=y−4\frac{4y(y - 4)}{4y} = y - 44y4y(y−4)=y−4
63. Simplify:
2z2+8z2z\frac{2z^2 + 8z}{2z}2z2z2+8z
Factor the numerator:
2z(z+4)2z=z+4\frac{2z(z + 4)}{2z} = z + 42z2z(z+4)=z+4
64. Expand:
(2x−3)2(2x - 3)^2(2x−3)2
Use the binomial expansion formula (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab +
b^2(a−b)2=a2−2ab+b2:
(2x−3)2=4x2−12x+9(2x - 3)^2 = 4x^2 - 12x + 9(2x−3)2=4x2−12x+9
65. Expand:
(y+4)2(y + 4)^2(y+4)2
Use the binomial expansion formula (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab +
b^2(a+b)2=a2+2ab+b2:
(y+4)2=y2+8y+16(y + 4)^2 = y^2 + 8y + 16(y+4)2=y2+8y+16
66. Expand:
(3a−2b)(3a+2b)(3a - 2b)(3a + 2b)(3a−2b)(3a+2b)
Use the difference of squares formula (a−b)(a+b)=a2−b2(a - b)(a + b) = a^2 - b^2(a−b)
(a+b)=a2−b2:
(3a−2b)(3a+2b)=9a2−4b2(3a - 2b)(3a + 2b) = 9a^2 - 4b^2(3a−2b)(3a+2b)=9a2−4b2
67. Factor:
x2−9x^2 - 9x2−9
This is a difference of squares:
5. Solving the equation:
3z−14+z+23=5\{3z-1}{4} + \{z+2}{3} = 543z−1+3z+2=5
Find a common denominator (12):
3(3z−1)12+4(z+2)12=59z−3+4z+812=513z+512=513z+5=6013z=55z=5513≈4.23\{3(3z-1)}
{12} + \{4(z+2)}{12} = 5 \\ \{9z - 3 + 4z + 8}{12} = 5 \\ \{13z + 5}{12} = 5 \\ 13z + 5 =
60 \\ 13z = 55 \\ z = \{55}{13} \approx 4.23123(3z−1)+124(z+2)=5129z−3+4z+8=51213z+5
=513z+5=6013z=55z=1355≈4.23
6. Solving the equation:
2(a−3)+3(a+4)=52(a-3) + 3(a+4) = 52(a−3)+3(a+4)=5
Expand and combine like terms:
2a−6+3a+12=55a+6=55a=−1a=−152a - 6 + 3a + 12 = 5 \\ 5a + 6 = 5 \\ 5a = -1 \\ a = -\{1}
{5}2a−6+3a+12=55a+6=55a=−1a=−51
7. Solving the equation:
5(b+2)−2(b−3)=105(b+2) - 2(b-3) = 105(b+2)−2(b−3)=10
Expand and combine like terms:
5b+10−2b+6=103b+16=103b=−6b=−25b + 10 - 2b + 6 = 10 \\ 3b + 16 = 10 \\ 3b = -6 \\ b = -
25b+10−2b+6=103b+16=103b=−6b=−2
8. Solving the equation:
(x+1)2−4(x−2)=0(x+1)^2 - 4(x-2) = 0(x+1)2−4(x−2)=0
Expand and simplify:
(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0(x+1)^2 - 4x + 8 = 0 \\ x^2 + 2x + 1 - 4x + 8 =
0 \\ x^2 - 2x + 9 = 0(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0
Solve using the quadratic formula x=−b±b2−4ac2ax = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}x=2a−b±b2−4ac:
x=2±4−362x=2±−322x=1±i8x = \{2 \pm \sqrt{4 - 36}}{2} \\ x = \{2 \pm \sqrt{-32}}{2} \\ x
= 1 \pm i\sqrt{8}x=22±4−36x=22±−32x=1±i8
So, x=1+22ix = 1 + 2\sqrt{2}ix=1+22i and x=1−22ix = 1 - 2\sqrt{2}ix=1−22i.
9. Solving the equation:
3(y−1)2+2(y+3)=03(y-1)^2 + 2(y+3) = 03(y−1)2+2(y+3)=0
Expand and combine like terms:
3(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=03(y^2 - 2y + 1) + 2y + 6 = 0 \\ 3y^2 - 6y
+ 3 + 2y + 6 = 0 \\ 3y^2 - 4y + 9 = 03(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=0
Solve using the quadratic formula y=−b±b2−4ac2ay = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}y=2a−b±b2−4ac:
y=4±16−1086y=4±−926y=4±2i236y=2±i233y = \{4 \pm \sqrt{16 - 108}}{6} \\ y = \{4 \pm \
sqrt{-92}}{6} \\ y = \{4 \pm 2i\sqrt{23}}{6} \\ y = \{2 \pm i\sqrt{23}}{3}y=64±16−108
y=64±−92y=64±2i23y=32±i23
So, y=2+i233y = \{2 + i\sqrt{23}}{3}y=32+i23 and y=2−i233y = \{2 - i\sqrt{23}}
{3}y=32−i23.
10. Solving the equation:
(z−2)2−5(z+1)=0(z-2)^2 - 5(z+1) = 0(z−2)2−5(z+1)=0
Expand and simplify:
z2−4z+4−5z−5=0z2−9z−1=0z^2 - 4z + 4 - 5z - 5 = 0 \\ z^2 - 9z - 1 =
0z2−4z+4−5z−5=0z2−9z−1=0
Solve using the quadratic formula z=−b±b2−4ac2az = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}z=2a−b±b2−4ac:
z=9±81+42z=9±852z = \{9 \pm \sqrt{81 + 4}}{2} \\ z = \{9 \pm \sqrt{85}}{2}z=29±81+4
z=29±85
So, z=9+852z = \{9 + \sqrt{85}}{2}z=29+85 and z=9−852z = \{9 - \sqrt{85}}{2}z=29−85.
Advanced Algebraic Expressions
61. Simplify:
3x2−9x3x\frac{3x^2 - 9x}{3x}3x3x2−9x
Factor the numerator:
3x(x−3)3x=x−3\frac{3x(x - 3)}{3x} = x - 33x3x(x−3)=x−3
62. Simplify:
4y2−16y4y\frac{4y^2 - 16y}{4y}4y4y2−16y
Factor the numerator:
4y(y−4)4y=y−4\frac{4y(y - 4)}{4y} = y - 44y4y(y−4)=y−4
63. Simplify:
2z2+8z2z\frac{2z^2 + 8z}{2z}2z2z2+8z
Factor the numerator:
2z(z+4)2z=z+4\frac{2z(z + 4)}{2z} = z + 42z2z(z+4)=z+4
64. Expand:
(2x−3)2(2x - 3)^2(2x−3)2
Use the binomial expansion formula (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab +
b^2(a−b)2=a2−2ab+b2:
(2x−3)2=4x2−12x+9(2x - 3)^2 = 4x^2 - 12x + 9(2x−3)2=4x2−12x+9
65. Expand:
(y+4)2(y + 4)^2(y+4)2
Use the binomial expansion formula (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab +
b^2(a+b)2=a2+2ab+b2:
(y+4)2=y2+8y+16(y + 4)^2 = y^2 + 8y + 16(y+4)2=y2+8y+16
66. Expand:
(3a−2b)(3a+2b)(3a - 2b)(3a + 2b)(3a−2b)(3a+2b)
Use the difference of squares formula (a−b)(a+b)=a2−b2(a - b)(a + b) = a^2 - b^2(a−b)
(a+b)=a2−b2:
(3a−2b)(3a+2b)=9a2−4b2(3a - 2b)(3a + 2b) = 9a^2 - 4b^2(3a−2b)(3a+2b)=9a2−4b2
67. Factor:
x2−9x^2 - 9x2−9
This is a difference of squares:
5. Solving the equation:
3z−14+z+23=5\{3z-1}{4} + \{z+2}{3} = 543z−1+3z+2=5
Find a common denominator (12):
3(3z−1)12+4(z+2)12=59z−3+4z+812=513z+512=513z+5=6013z=55z=5513≈4.23\{3(3z-1)}
{12} + \{4(z+2)}{12} = 5 \\ \{9z - 3 + 4z + 8}{12} = 5 \\ \{13z + 5}{12} = 5 \\ 13z + 5 =
60 \\ 13z = 55 \\ z = \{55}{13} \approx 4.23123(3z−1)+124(z+2)=5129z−3+4z+8=51213z+5
=513z+5=6013z=55z=1355≈4.23
6. Solving the equation:
2(a−3)+3(a+4)=52(a-3) + 3(a+4) = 52(a−3)+3(a+4)=5
Expand and combine like terms:
2a−6+3a+12=55a+6=55a=−1a=−152a - 6 + 3a + 12 = 5 \\ 5a + 6 = 5 \\ 5a = -1 \\ a = -\{1}
{5}2a−6+3a+12=55a+6=55a=−1a=−51
7. Solving the equation:
5(b+2)−2(b−3)=105(b+2) - 2(b-3) = 105(b+2)−2(b−3)=10
Expand and combine like terms:
5b+10−2b+6=103b+16=103b=−6b=−25b + 10 - 2b + 6 = 10 \\ 3b + 16 = 10 \\ 3b = -6 \\ b = -
25b+10−2b+6=103b+16=103b=−6b=−2
8. Solving the equation:
(x+1)2−4(x−2)=0(x+1)^2 - 4(x-2) = 0(x+1)2−4(x−2)=0
Expand and simplify:
(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0(x+1)^2 - 4x + 8 = 0 \\ x^2 + 2x + 1 - 4x + 8 =
0 \\ x^2 - 2x + 9 = 0(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0
Solve using the quadratic formula x=−b±b2−4ac2ax = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}x=2a−b±b2−4ac:
x=2±4−362x=2±−322x=1±i8x = \{2 \pm \sqrt{4 - 36}}{2} \\ x = \{2 \pm \sqrt{-32}}{2} \\ x
= 1 \pm i\sqrt{8}x=22±4−36x=22±−32x=1±i8
So, x=1+22ix = 1 + 2\sqrt{2}ix=1+22i and x=1−22ix = 1 - 2\sqrt{2}ix=1−22i.
9. Solving the equation:
3(y−1)2+2(y+3)=03(y-1)^2 + 2(y+3) = 03(y−1)2+2(y+3)=0
Expand and combine like terms:
3(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=03(y^2 - 2y + 1) + 2y + 6 = 0 \\ 3y^2 - 6y
+ 3 + 2y + 6 = 0 \\ 3y^2 - 4y + 9 = 03(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=0
Solve using the quadratic formula y=−b±b2−4ac2ay = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}y=2a−b±b2−4ac:
y=4±16−1086y=4±−926y=4±2i236y=2±i233y = \{4 \pm \sqrt{16 - 108}}{6} \\ y = \{4 \pm \
sqrt{-92}}{6} \\ y = \{4 \pm 2i\sqrt{23}}{6} \\ y = \{2 \pm i\sqrt{23}}{3}y=64±16−108
y=64±−92y=64±2i23y=32±i23
So, y=2+i233y = \{2 + i\sqrt{23}}{3}y=32+i23 and y=2−i233y = \{2 - i\sqrt{23}}
{3}y=32−i23.
10. Solving the equation:
(z−2)2−5(z+1)=0(z-2)^2 - 5(z+1) = 0(z−2)2−5(z+1)=0
Expand and simplify:
z2−4z+4−5z−5=0z2−9z−1=0z^2 - 4z + 4 - 5z - 5 = 0 \\ z^2 - 9z - 1 =
0z2−4z+4−5z−5=0z2−9z−1=0
Solve using the quadratic formula z=−b±b2−4ac2az = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}z=2a−b±b2−4ac:
z=9±81+42z=9±852z = \{9 \pm \sqrt{81 + 4}}{2} \\ z = \{9 \pm \sqrt{85}}{2}z=29±81+4
z=29±85
So, z=9+852z = \{9 + \sqrt{85}}{2}z=29+85 and z=9−852z = \{9 - \sqrt{85}}{2}z=29−85.
Advanced Algebraic Expressions
61. Simplify:
3x2−9x3x\frac{3x^2 - 9x}{3x}3x3x2−9x
Factor the numerator:
3x(x−3)3x=x−3\frac{3x(x - 3)}{3x} = x - 33x3x(x−3)=x−3
62. Simplify:
4y2−16y4y\frac{4y^2 - 16y}{4y}4y4y2−16y
Factor the numerator:
4y(y−4)4y=y−4\frac{4y(y - 4)}{4y} = y - 44y4y(y−4)=y−4
63. Simplify:
2z2+8z2z\frac{2z^2 + 8z}{2z}2z2z2+8z
Factor the numerator:
2z(z+4)2z=z+4\frac{2z(z + 4)}{2z} = z + 42z2z(z+4)=z+4
64. Expand:
(2x−3)2(2x - 3)^2(2x−3)2
Use the binomial expansion formula (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab +
b^2(a−b)2=a2−2ab+b2:
(2x−3)2=4x2−12x+9(2x - 3)^2 = 4x^2 - 12x + 9(2x−3)2=4x2−12x+9
65. Expand:
(y+4)2(y + 4)^2(y+4)2
Use the binomial expansion formula (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab +
b^2(a+b)2=a2+2ab+b2:
(y+4)2=y2+8y+16(y + 4)^2 = y^2 + 8y + 16(y+4)2=y2+8y+16
66. Expand:
(3a−2b)(3a+2b)(3a - 2b)(3a + 2b)(3a−2b)(3a+2b)
Use the difference of squares formula (a−b)(a+b)=a2−b2(a - b)(a + b) = a^2 - b^2(a−b)
(a+b)=a2−b2:
(3a−2b)(3a+2b)=9a2−4b2(3a - 2b)(3a + 2b) = 9a^2 - 4b^2(3a−2b)(3a+2b)=9a2−4b2
67. Factor:
x2−9x^2 - 9x2−9
This is a difference of squares:
5. Solving the equation:
3z−14+z+23=5\{3z-1}{4} + \{z+2}{3} = 543z−1+3z+2=5
Find a common denominator (12):
3(3z−1)12+4(z+2)12=59z−3+4z+812=513z+512=513z+5=6013z=55z=5513≈4.23\{3(3z-1)}
{12} + \{4(z+2)}{12} = 5 \\ \{9z - 3 + 4z + 8}{12} = 5 \\ \{13z + 5}{12} = 5 \\ 13z + 5 =
60 \\ 13z = 55 \\ z = \{55}{13} \approx 4.23123(3z−1)+124(z+2)=5129z−3+4z+8=51213z+5
=513z+5=6013z=55z=1355≈4.23
6. Solving the equation:
2(a−3)+3(a+4)=52(a-3) + 3(a+4) = 52(a−3)+3(a+4)=5
Expand and combine like terms:
2a−6+3a+12=55a+6=55a=−1a=−152a - 6 + 3a + 12 = 5 \\ 5a + 6 = 5 \\ 5a = -1 \\ a = -\{1}
{5}2a−6+3a+12=55a+6=55a=−1a=−51
7. Solving the equation:
5(b+2)−2(b−3)=105(b+2) - 2(b-3) = 105(b+2)−2(b−3)=10
Expand and combine like terms:
5b+10−2b+6=103b+16=103b=−6b=−25b + 10 - 2b + 6 = 10 \\ 3b + 16 = 10 \\ 3b = -6 \\ b = -
25b+10−2b+6=103b+16=103b=−6b=−2
8. Solving the equation:
(x+1)2−4(x−2)=0(x+1)^2 - 4(x-2) = 0(x+1)2−4(x−2)=0
Expand and simplify:
(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0(x+1)^2 - 4x + 8 = 0 \\ x^2 + 2x + 1 - 4x + 8 =
0 \\ x^2 - 2x + 9 = 0(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0
Solve using the quadratic formula x=−b±b2−4ac2ax = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}x=2a−b±b2−4ac:
x=2±4−362x=2±−322x=1±i8x = \{2 \pm \sqrt{4 - 36}}{2} \\ x = \{2 \pm \sqrt{-32}}{2} \\ x
= 1 \pm i\sqrt{8}x=22±4−36x=22±−32x=1±i8
So, x=1+22ix = 1 + 2\sqrt{2}ix=1+22i and x=1−22ix = 1 - 2\sqrt{2}ix=1−22i.
9. Solving the equation:
3(y−1)2+2(y+3)=03(y-1)^2 + 2(y+3) = 03(y−1)2+2(y+3)=0
Expand and combine like terms:
3(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=03(y^2 - 2y + 1) + 2y + 6 = 0 \\ 3y^2 - 6y
+ 3 + 2y + 6 = 0 \\ 3y^2 - 4y + 9 = 03(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=0
Solve using the quadratic formula y=−b±b2−4ac2ay = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}y=2a−b±b2−4ac:
y=4±16−1086y=4±−926y=4±2i236y=2±i233y = \{4 \pm \sqrt{16 - 108}}{6} \\ y = \{4 \pm \
sqrt{-92}}{6} \\ y = \{4 \pm 2i\sqrt{23}}{6} \\ y = \{2 \pm i\sqrt{23}}{3}y=64±16−108
y=64±−92y=64±2i23y=32±i23
So, y=2+i233y = \{2 + i\sqrt{23}}{3}y=32+i23 and y=2−i233y = \{2 - i\sqrt{23}}
{3}y=32−i23.
10. Solving the equation:
(z−2)2−5(z+1)=0(z-2)^2 - 5(z+1) = 0(z−2)2−5(z+1)=0
Expand and simplify:
z2−4z+4−5z−5=0z2−9z−1=0z^2 - 4z + 4 - 5z - 5 = 0 \\ z^2 - 9z - 1 =
0z2−4z+4−5z−5=0z2−9z−1=0
Solve using the quadratic formula z=−b±b2−4ac2az = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}z=2a−b±b2−4ac:
z=9±81+42z=9±852z = \{9 \pm \sqrt{81 + 4}}{2} \\ z = \{9 \pm \sqrt{85}}{2}z=29±81+4
z=29±85
So, z=9+852z = \{9 + \sqrt{85}}{2}z=29+85 and z=9−852z = \{9 - \sqrt{85}}{2}z=29−85.
Advanced Algebraic Expressions
61. Simplify:
3x2−9x3x\frac{3x^2 - 9x}{3x}3x3x2−9x
Factor the numerator:
3x(x−3)3x=x−3\frac{3x(x - 3)}{3x} = x - 33x3x(x−3)=x−3
62. Simplify:
4y2−16y4y\frac{4y^2 - 16y}{4y}4y4y2−16y
Factor the numerator:
4y(y−4)4y=y−4\frac{4y(y - 4)}{4y} = y - 44y4y(y−4)=y−4
63. Simplify:
2z2+8z2z\frac{2z^2 + 8z}{2z}2z2z2+8z
Factor the numerator:
2z(z+4)2z=z+4\frac{2z(z + 4)}{2z} = z + 42z2z(z+4)=z+4
64. Expand:
(2x−3)2(2x - 3)^2(2x−3)2
Use the binomial expansion formula (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab +
b^2(a−b)2=a2−2ab+b2:
(2x−3)2=4x2−12x+9(2x - 3)^2 = 4x^2 - 12x + 9(2x−3)2=4x2−12x+9
65. Expand:
(y+4)2(y + 4)^2(y+4)2
Use the binomial expansion formula (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab +
b^2(a+b)2=a2+2ab+b2:
(y+4)2=y2+8y+16(y + 4)^2 = y^2 + 8y + 16(y+4)2=y2+8y+16
66. Expand:
(3a−2b)(3a+2b)(3a - 2b)(3a + 2b)(3a−2b)(3a+2b)
Use the difference of squares formula (a−b)(a+b)=a2−b2(a - b)(a + b) = a^2 - b^2(a−b)
(a+b)=a2−b2:
(3a−2b)(3a+2b)=9a2−4b2(3a - 2b)(3a + 2b) = 9a^2 - 4b^2(3a−2b)(3a+2b)=9a2−4b2
67. Factor:
x2−9x^2 - 9x2−9
This is a difference of squares:
5. Solving the equation:
3z−14+z+23=5\{3z-1}{4} + \{z+2}{3} = 543z−1+3z+2=5
Find a common denominator (12):
3(3z−1)12+4(z+2)12=59z−3+4z+812=513z+512=513z+5=6013z=55z=5513≈4.23\{3(3z-1)}
{12} + \{4(z+2)}{12} = 5 \\ \{9z - 3 + 4z + 8}{12} = 5 \\ \{13z + 5}{12} = 5 \\ 13z + 5 =
60 \\ 13z = 55 \\ z = \{55}{13} \approx 4.23123(3z−1)+124(z+2)=5129z−3+4z+8=51213z+5
=513z+5=6013z=55z=1355≈4.23
6. Solving the equation:
2(a−3)+3(a+4)=52(a-3) + 3(a+4) = 52(a−3)+3(a+4)=5
Expand and combine like terms:
2a−6+3a+12=55a+6=55a=−1a=−152a - 6 + 3a + 12 = 5 \\ 5a + 6 = 5 \\ 5a = -1 \\ a = -\{1}
{5}2a−6+3a+12=55a+6=55a=−1a=−51
7. Solving the equation:
5(b+2)−2(b−3)=105(b+2) - 2(b-3) = 105(b+2)−2(b−3)=10
Expand and combine like terms:
5b+10−2b+6=103b+16=103b=−6b=−25b + 10 - 2b + 6 = 10 \\ 3b + 16 = 10 \\ 3b = -6 \\ b = -
25b+10−2b+6=103b+16=103b=−6b=−2
8. Solving the equation:
(x+1)2−4(x−2)=0(x+1)^2 - 4(x-2) = 0(x+1)2−4(x−2)=0
Expand and simplify:
(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0(x+1)^2 - 4x + 8 = 0 \\ x^2 + 2x + 1 - 4x + 8 =
0 \\ x^2 - 2x + 9 = 0(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0
Solve using the quadratic formula x=−b±b2−4ac2ax = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}x=2a−b±b2−4ac:
x=2±4−362x=2±−322x=1±i8x = \{2 \pm \sqrt{4 - 36}}{2} \\ x = \{2 \pm \sqrt{-32}}{2} \\ x
= 1 \pm i\sqrt{8}x=22±4−36x=22±−32x=1±i8
So, x=1+22ix = 1 + 2\sqrt{2}ix=1+22i and x=1−22ix = 1 - 2\sqrt{2}ix=1−22i.
9. Solving the equation:
3(y−1)2+2(y+3)=03(y-1)^2 + 2(y+3) = 03(y−1)2+2(y+3)=0
Expand and combine like terms:
3(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=03(y^2 - 2y + 1) + 2y + 6 = 0 \\ 3y^2 - 6y
+ 3 + 2y + 6 = 0 \\ 3y^2 - 4y + 9 = 03(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=0
Solve using the quadratic formula y=−b±b2−4ac2ay = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}y=2a−b±b2−4ac:
y=4±16−1086y=4±−926y=4±2i236y=2±i233y = \{4 \pm \sqrt{16 - 108}}{6} \\ y = \{4 \pm \
sqrt{-92}}{6} \\ y = \{4 \pm 2i\sqrt{23}}{6} \\ y = \{2 \pm i\sqrt{23}}{3}y=64±16−108
y=64±−92y=64±2i23y=32±i23
So, y=2+i233y = \{2 + i\sqrt{23}}{3}y=32+i23 and y=2−i233y = \{2 - i\sqrt{23}}
{3}y=32−i23.
10. Solving the equation:
(z−2)2−5(z+1)=0(z-2)^2 - 5(z+1) = 0(z−2)2−5(z+1)=0
Expand and simplify:
z2−4z+4−5z−5=0z2−9z−1=0z^2 - 4z + 4 - 5z - 5 = 0 \\ z^2 - 9z - 1 =
0z2−4z+4−5z−5=0z2−9z−1=0
Solve using the quadratic formula z=−b±b2−4ac2az = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}z=2a−b±b2−4ac:
z=9±81+42z=9±852z = \{9 \pm \sqrt{81 + 4}}{2} \\ z = \{9 \pm \sqrt{85}}{2}z=29±81+4
z=29±85
So, z=9+852z = \{9 + \sqrt{85}}{2}z=29+85 and z=9−852z = \{9 - \sqrt{85}}{2}z=29−85.
Advanced Algebraic Expressions
61. Simplify:
3x2−9x3x\frac{3x^2 - 9x}{3x}3x3x2−9x
Factor the numerator:
3x(x−3)3x=x−3\frac{3x(x - 3)}{3x} = x - 33x3x(x−3)=x−3
62. Simplify:
4y2−16y4y\frac{4y^2 - 16y}{4y}4y4y2−16y
Factor the numerator:
4y(y−4)4y=y−4\frac{4y(y - 4)}{4y} = y - 44y4y(y−4)=y−4
63. Simplify:
2z2+8z2z\frac{2z^2 + 8z}{2z}2z2z2+8z
Factor the numerator:
2z(z+4)2z=z+4\frac{2z(z + 4)}{2z} = z + 42z2z(z+4)=z+4
64. Expand:
(2x−3)2(2x - 3)^2(2x−3)2
Use the binomial expansion formula (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab +
b^2(a−b)2=a2−2ab+b2:
(2x−3)2=4x2−12x+9(2x - 3)^2 = 4x^2 - 12x + 9(2x−3)2=4x2−12x+9
65. Expand:
(y+4)2(y + 4)^2(y+4)2
Use the binomial expansion formula (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab +
b^2(a+b)2=a2+2ab+b2:
(y+4)2=y2+8y+16(y + 4)^2 = y^2 + 8y + 16(y+4)2=y2+8y+16
66. Expand:
(3a−2b)(3a+2b)(3a - 2b)(3a + 2b)(3a−2b)(3a+2b)
Use the difference of squares formula (a−b)(a+b)=a2−b2(a - b)(a + b) = a^2 - b^2(a−b)
(a+b)=a2−b2:
(3a−2b)(3a+2b)=9a2−4b2(3a - 2b)(3a + 2b) = 9a^2 - 4b^2(3a−2b)(3a+2b)=9a2−4b2
67. Factor:
x2−9x^2 - 9x2−9
This is a difference of squares:
5. Solving the equation:
3z−14+z+23=5\{3z-1}{4} + \{z+2}{3} = 543z−1+3z+2=5
Find a common denominator (12):
3(3z−1)12+4(z+2)12=59z−3+4z+812=513z+512=513z+5=6013z=55z=5513≈4.23\{3(3z-1)}
{12} + \{4(z+2)}{12} = 5 \\ \{9z - 3 + 4z + 8}{12} = 5 \\ \{13z + 5}{12} = 5 \\ 13z + 5 =
60 \\ 13z = 55 \\ z = \{55}{13} \approx 4.23123(3z−1)+124(z+2)=5129z−3+4z+8=51213z+5
=513z+5=6013z=55z=1355≈4.23
6. Solving the equation:
2(a−3)+3(a+4)=52(a-3) + 3(a+4) = 52(a−3)+3(a+4)=5
Expand and combine like terms:
2a−6+3a+12=55a+6=55a=−1a=−152a - 6 + 3a + 12 = 5 \\ 5a + 6 = 5 \\ 5a = -1 \\ a = -\{1}
{5}2a−6+3a+12=55a+6=55a=−1a=−51
7. Solving the equation:
5(b+2)−2(b−3)=105(b+2) - 2(b-3) = 105(b+2)−2(b−3)=10
Expand and combine like terms:
5b+10−2b+6=103b+16=103b=−6b=−25b + 10 - 2b + 6 = 10 \\ 3b + 16 = 10 \\ 3b = -6 \\ b = -
25b+10−2b+6=103b+16=103b=−6b=−2
8. Solving the equation:
(x+1)2−4(x−2)=0(x+1)^2 - 4(x-2) = 0(x+1)2−4(x−2)=0
Expand and simplify:
(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0(x+1)^2 - 4x + 8 = 0 \\ x^2 + 2x + 1 - 4x + 8 =
0 \\ x^2 - 2x + 9 = 0(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0
Solve using the quadratic formula x=−b±b2−4ac2ax = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}x=2a−b±b2−4ac:
x=2±4−362x=2±−322x=1±i8x = \{2 \pm \sqrt{4 - 36}}{2} \\ x = \{2 \pm \sqrt{-32}}{2} \\ x
= 1 \pm i\sqrt{8}x=22±4−36x=22±−32x=1±i8
So, x=1+22ix = 1 + 2\sqrt{2}ix=1+22i and x=1−22ix = 1 - 2\sqrt{2}ix=1−22i.
9. Solving the equation:
3(y−1)2+2(y+3)=03(y-1)^2 + 2(y+3) = 03(y−1)2+2(y+3)=0
Expand and combine like terms:
3(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=03(y^2 - 2y + 1) + 2y + 6 = 0 \\ 3y^2 - 6y
+ 3 + 2y + 6 = 0 \\ 3y^2 - 4y + 9 = 03(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=0
Solve using the quadratic formula y=−b±b2−4ac2ay = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}y=2a−b±b2−4ac:
y=4±16−1086y=4±−926y=4±2i236y=2±i233y = \{4 \pm \sqrt{16 - 108}}{6} \\ y = \{4 \pm \
sqrt{-92}}{6} \\ y = \{4 \pm 2i\sqrt{23}}{6} \\ y = \{2 \pm i\sqrt{23}}{3}y=64±16−108
y=64±−92y=64±2i23y=32±i23
So, y=2+i233y = \{2 + i\sqrt{23}}{3}y=32+i23 and y=2−i233y = \{2 - i\sqrt{23}}
{3}y=32−i23.
10. Solving the equation:
(z−2)2−5(z+1)=0(z-2)^2 - 5(z+1) = 0(z−2)2−5(z+1)=0
Expand and simplify:
z2−4z+4−5z−5=0z2−9z−1=0z^2 - 4z + 4 - 5z - 5 = 0 \\ z^2 - 9z - 1 =
0z2−4z+4−5z−5=0z2−9z−1=0
Solve using the quadratic formula z=−b±b2−4ac2az = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}z=2a−b±b2−4ac:
z=9±81+42z=9±852z = \{9 \pm \sqrt{81 + 4}}{2} \\ z = \{9 \pm \sqrt{85}}{2}z=29±81+4
z=29±85
So, z=9+852z = \{9 + \sqrt{85}}{2}z=29+85 and z=9−852z = \{9 - \sqrt{85}}{2}z=29−85.
Advanced Algebraic Expressions
61. Simplify:
3x2−9x3x\frac{3x^2 - 9x}{3x}3x3x2−9x
Factor the numerator:
3x(x−3)3x=x−3\frac{3x(x - 3)}{3x} = x - 33x3x(x−3)=x−3
62. Simplify:
4y2−16y4y\frac{4y^2 - 16y}{4y}4y4y2−16y
Factor the numerator:
4y(y−4)4y=y−4\frac{4y(y - 4)}{4y} = y - 44y4y(y−4)=y−4
63. Simplify:
2z2+8z2z\frac{2z^2 + 8z}{2z}2z2z2+8z
Factor the numerator:
2z(z+4)2z=z+4\frac{2z(z + 4)}{2z} = z + 42z2z(z+4)=z+4
64. Expand:
(2x−3)2(2x - 3)^2(2x−3)2
Use the binomial expansion formula (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab +
b^2(a−b)2=a2−2ab+b2:
(2x−3)2=4x2−12x+9(2x - 3)^2 = 4x^2 - 12x + 9(2x−3)2=4x2−12x+9
65. Expand:
(y+4)2(y + 4)^2(y+4)2
Use the binomial expansion formula (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab +
b^2(a+b)2=a2+2ab+b2:
(y+4)2=y2+8y+16(y + 4)^2 = y^2 + 8y + 16(y+4)2=y2+8y+16
66. Expand:
(3a−2b)(3a+2b)(3a - 2b)(3a + 2b)(3a−2b)(3a+2b)
Use the difference of squares formula (a−b)(a+b)=a2−b2(a - b)(a + b) = a^2 - b^2(a−b)
(a+b)=a2−b2:
(3a−2b)(3a+2b)=9a2−4b2(3a - 2b)(3a + 2b) = 9a^2 - 4b^2(3a−2b)(3a+2b)=9a2−4b2
67. Factor:
x2−9x^2 - 9x2−9
This is a difference of squares:
5. Solving the equation:
3z−14+z+23=5\{3z-1}{4} + \{z+2}{3} = 543z−1+3z+2=5
Find a common denominator (12):
3(3z−1)12+4(z+2)12=59z−3+4z+812=513z+512=513z+5=6013z=55z=5513≈4.23\{3(3z-1)}
{12} + \{4(z+2)}{12} = 5 \\ \{9z - 3 + 4z + 8}{12} = 5 \\ \{13z + 5}{12} = 5 \\ 13z + 5 =
60 \\ 13z = 55 \\ z = \{55}{13} \approx 4.23123(3z−1)+124(z+2)=5129z−3+4z+8=51213z+5
=513z+5=6013z=55z=1355≈4.23
6. Solving the equation:
2(a−3)+3(a+4)=52(a-3) + 3(a+4) = 52(a−3)+3(a+4)=5
Expand and combine like terms:
2a−6+3a+12=55a+6=55a=−1a=−152a - 6 + 3a + 12 = 5 \\ 5a + 6 = 5 \\ 5a = -1 \\ a = -\{1}
{5}2a−6+3a+12=55a+6=55a=−1a=−51
7. Solving the equation:
5(b+2)−2(b−3)=105(b+2) - 2(b-3) = 105(b+2)−2(b−3)=10
Expand and combine like terms:
5b+10−2b+6=103b+16=103b=−6b=−25b + 10 - 2b + 6 = 10 \\ 3b + 16 = 10 \\ 3b = -6 \\ b = -
25b+10−2b+6=103b+16=103b=−6b=−2
8. Solving the equation:
(x+1)2−4(x−2)=0(x+1)^2 - 4(x-2) = 0(x+1)2−4(x−2)=0
Expand and simplify:
(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0(x+1)^2 - 4x + 8 = 0 \\ x^2 + 2x + 1 - 4x + 8 =
0 \\ x^2 - 2x + 9 = 0(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0
Solve using the quadratic formula x=−b±b2−4ac2ax = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}x=2a−b±b2−4ac:
x=2±4−362x=2±−322x=1±i8x = \{2 \pm \sqrt{4 - 36}}{2} \\ x = \{2 \pm \sqrt{-32}}{2} \\ x
= 1 \pm i\sqrt{8}x=22±4−36x=22±−32x=1±i8
So, x=1+22ix = 1 + 2\sqrt{2}ix=1+22i and x=1−22ix = 1 - 2\sqrt{2}ix=1−22i.
9. Solving the equation:
3(y−1)2+2(y+3)=03(y-1)^2 + 2(y+3) = 03(y−1)2+2(y+3)=0
Expand and combine like terms:
3(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=03(y^2 - 2y + 1) + 2y + 6 = 0 \\ 3y^2 - 6y
+ 3 + 2y + 6 = 0 \\ 3y^2 - 4y + 9 = 03(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=0
Solve using the quadratic formula y=−b±b2−4ac2ay = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}y=2a−b±b2−4ac:
y=4±16−1086y=4±−926y=4±2i236y=2±i233y = \{4 \pm \sqrt{16 - 108}}{6} \\ y = \{4 \pm \
sqrt{-92}}{6} \\ y = \{4 \pm 2i\sqrt{23}}{6} \\ y = \{2 \pm i\sqrt{23}}{3}y=64±16−108
y=64±−92y=64±2i23y=32±i23
So, y=2+i233y = \{2 + i\sqrt{23}}{3}y=32+i23 and y=2−i233y = \{2 - i\sqrt{23}}
{3}y=32−i23.
10. Solving the equation:
(z−2)2−5(z+1)=0(z-2)^2 - 5(z+1) = 0(z−2)2−5(z+1)=0
Expand and simplify:
z2−4z+4−5z−5=0z2−9z−1=0z^2 - 4z + 4 - 5z - 5 = 0 \\ z^2 - 9z - 1 =
0z2−4z+4−5z−5=0z2−9z−1=0
Solve using the quadratic formula z=−b±b2−4ac2az = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}z=2a−b±b2−4ac:
z=9±81+42z=9±852z = \{9 \pm \sqrt{81 + 4}}{2} \\ z = \{9 \pm \sqrt{85}}{2}z=29±81+4
z=29±85
So, z=9+852z = \{9 + \sqrt{85}}{2}z=29+85 and z=9−852z = \{9 - \sqrt{85}}{2}z=29−85.
Advanced Algebraic Expressions
61. Simplify:
3x2−9x3x\frac{3x^2 - 9x}{3x}3x3x2−9x
Factor the numerator:
3x(x−3)3x=x−3\frac{3x(x - 3)}{3x} = x - 33x3x(x−3)=x−3
62. Simplify:
4y2−16y4y\frac{4y^2 - 16y}{4y}4y4y2−16y
Factor the numerator:
4y(y−4)4y=y−4\frac{4y(y - 4)}{4y} = y - 44y4y(y−4)=y−4
63. Simplify:
2z2+8z2z\frac{2z^2 + 8z}{2z}2z2z2+8z
Factor the numerator:
2z(z+4)2z=z+4\frac{2z(z + 4)}{2z} = z + 42z2z(z+4)=z+4
64. Expand:
(2x−3)2(2x - 3)^2(2x−3)2
Use the binomial expansion formula (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab +
b^2(a−b)2=a2−2ab+b2:
(2x−3)2=4x2−12x+9(2x - 3)^2 = 4x^2 - 12x + 9(2x−3)2=4x2−12x+9
65. Expand:
(y+4)2(y + 4)^2(y+4)2
Use the binomial expansion formula (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab +
b^2(a+b)2=a2+2ab+b2:
(y+4)2=y2+8y+16(y + 4)^2 = y^2 + 8y + 16(y+4)2=y2+8y+16
66. Expand:
(3a−2b)(3a+2b)(3a - 2b)(3a + 2b)(3a−2b)(3a+2b)
Use the difference of squares formula (a−b)(a+b)=a2−b2(a - b)(a + b) = a^2 - b^2(a−b)
(a+b)=a2−b2:
(3a−2b)(3a+2b)=9a2−4b2(3a - 2b)(3a + 2b) = 9a^2 - 4b^2(3a−2b)(3a+2b)=9a2−4b2
67. Factor:
x2−9x^2 - 9x2−9
This is a difference of squares:
5. Solving the equation:
3z−14+z+23=5\{3z-1}{4} + \{z+2}{3} = 543z−1+3z+2=5
Find a common denominator (12):
3(3z−1)12+4(z+2)12=59z−3+4z+812=513z+512=513z+5=6013z=55z=5513≈4.23\{3(3z-1)}
{12} + \{4(z+2)}{12} = 5 \\ \{9z - 3 + 4z + 8}{12} = 5 \\ \{13z + 5}{12} = 5 \\ 13z + 5 =
60 \\ 13z = 55 \\ z = \{55}{13} \approx 4.23123(3z−1)+124(z+2)=5129z−3+4z+8=51213z+5
=513z+5=6013z=55z=1355≈4.23
6. Solving the equation:
2(a−3)+3(a+4)=52(a-3) + 3(a+4) = 52(a−3)+3(a+4)=5
Expand and combine like terms:
2a−6+3a+12=55a+6=55a=−1a=−152a - 6 + 3a + 12 = 5 \\ 5a + 6 = 5 \\ 5a = -1 \\ a = -\{1}
{5}2a−6+3a+12=55a+6=55a=−1a=−51
7. Solving the equation:
5(b+2)−2(b−3)=105(b+2) - 2(b-3) = 105(b+2)−2(b−3)=10
Expand and combine like terms:
5b+10−2b+6=103b+16=103b=−6b=−25b + 10 - 2b + 6 = 10 \\ 3b + 16 = 10 \\ 3b = -6 \\ b = -
25b+10−2b+6=103b+16=103b=−6b=−2
8. Solving the equation:
(x+1)2−4(x−2)=0(x+1)^2 - 4(x-2) = 0(x+1)2−4(x−2)=0
Expand and simplify:
(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0(x+1)^2 - 4x + 8 = 0 \\ x^2 + 2x + 1 - 4x + 8 =
0 \\ x^2 - 2x + 9 = 0(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0
Solve using the quadratic formula x=−b±b2−4ac2ax = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}x=2a−b±b2−4ac:
x=2±4−362x=2±−322x=1±i8x = \{2 \pm \sqrt{4 - 36}}{2} \\ x = \{2 \pm \sqrt{-32}}{2} \\ x
= 1 \pm i\sqrt{8}x=22±4−36x=22±−32x=1±i8
So, x=1+22ix = 1 + 2\sqrt{2}ix=1+22i and x=1−22ix = 1 - 2\sqrt{2}ix=1−22i.
9. Solving the equation:
3(y−1)2+2(y+3)=03(y-1)^2 + 2(y+3) = 03(y−1)2+2(y+3)=0
Expand and combine like terms:
3(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=03(y^2 - 2y + 1) + 2y + 6 = 0 \\ 3y^2 - 6y
+ 3 + 2y + 6 = 0 \\ 3y^2 - 4y + 9 = 03(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=0
Solve using the quadratic formula y=−b±b2−4ac2ay = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}y=2a−b±b2−4ac:
y=4±16−1086y=4±−926y=4±2i236y=2±i233y = \{4 \pm \sqrt{16 - 108}}{6} \\ y = \{4 \pm \
sqrt{-92}}{6} \\ y = \{4 \pm 2i\sqrt{23}}{6} \\ y = \{2 \pm i\sqrt{23}}{3}y=64±16−108
y=64±−92y=64±2i23y=32±i23
So, y=2+i233y = \{2 + i\sqrt{23}}{3}y=32+i23 and y=2−i233y = \{2 - i\sqrt{23}}
{3}y=32−i23.
10. Solving the equation:
(z−2)2−5(z+1)=0(z-2)^2 - 5(z+1) = 0(z−2)2−5(z+1)=0
Expand and simplify:
z2−4z+4−5z−5=0z2−9z−1=0z^2 - 4z + 4 - 5z - 5 = 0 \\ z^2 - 9z - 1 =
0z2−4z+4−5z−5=0z2−9z−1=0
Solve using the quadratic formula z=−b±b2−4ac2az = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}z=2a−b±b2−4ac:
z=9±81+42z=9±852z = \{9 \pm \sqrt{81 + 4}}{2} \\ z = \{9 \pm \sqrt{85}}{2}z=29±81+4
z=29±85
So, z=9+852z = \{9 + \sqrt{85}}{2}z=29+85 and z=9−852z = \{9 - \sqrt{85}}{2}z=29−85.
Advanced Algebraic Expressions
61. Simplify:
3x2−9x3x\frac{3x^2 - 9x}{3x}3x3x2−9x
Factor the numerator:
3x(x−3)3x=x−3\frac{3x(x - 3)}{3x} = x - 33x3x(x−3)=x−3
62. Simplify:
4y2−16y4y\frac{4y^2 - 16y}{4y}4y4y2−16y
Factor the numerator:
4y(y−4)4y=y−4\frac{4y(y - 4)}{4y} = y - 44y4y(y−4)=y−4
63. Simplify:
2z2+8z2z\frac{2z^2 + 8z}{2z}2z2z2+8z
Factor the numerator:
2z(z+4)2z=z+4\frac{2z(z + 4)}{2z} = z + 42z2z(z+4)=z+4
64. Expand:
(2x−3)2(2x - 3)^2(2x−3)2
Use the binomial expansion formula (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab +
b^2(a−b)2=a2−2ab+b2:
(2x−3)2=4x2−12x+9(2x - 3)^2 = 4x^2 - 12x + 9(2x−3)2=4x2−12x+9
65. Expand:
(y+4)2(y + 4)^2(y+4)2
Use the binomial expansion formula (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab +
b^2(a+b)2=a2+2ab+b2:
(y+4)2=y2+8y+16(y + 4)^2 = y^2 + 8y + 16(y+4)2=y2+8y+16
66. Expand:
(3a−2b)(3a+2b)(3a - 2b)(3a + 2b)(3a−2b)(3a+2b)
Use the difference of squares formula (a−b)(a+b)=a2−b2(a - b)(a + b) = a^2 - b^2(a−b)
(a+b)=a2−b2:
(3a−2b)(3a+2b)=9a2−4b2(3a - 2b)(3a + 2b) = 9a^2 - 4b^2(3a−2b)(3a+2b)=9a2−4b2
67. Factor:
x2−9x^2 - 9x2−9
This is a difference of squares:
5. Solving the equation:
3z−14+z+23=5\{3z-1}{4} + \{z+2}{3} = 543z−1+3z+2=5
Find a common denominator (12):
3(3z−1)12+4(z+2)12=59z−3+4z+812=513z+512=513z+5=6013z=55z=5513≈4.23\{3(3z-1)}
{12} + \{4(z+2)}{12} = 5 \\ \{9z - 3 + 4z + 8}{12} = 5 \\ \{13z + 5}{12} = 5 \\ 13z + 5 =
60 \\ 13z = 55 \\ z = \{55}{13} \approx 4.23123(3z−1)+124(z+2)=5129z−3+4z+8=51213z+5
=513z+5=6013z=55z=1355≈4.23
6. Solving the equation:
2(a−3)+3(a+4)=52(a-3) + 3(a+4) = 52(a−3)+3(a+4)=5
Expand and combine like terms:
2a−6+3a+12=55a+6=55a=−1a=−152a - 6 + 3a + 12 = 5 \\ 5a + 6 = 5 \\ 5a = -1 \\ a = -\{1}
{5}2a−6+3a+12=55a+6=55a=−1a=−51
7. Solving the equation:
5(b+2)−2(b−3)=105(b+2) - 2(b-3) = 105(b+2)−2(b−3)=10
Expand and combine like terms:
5b+10−2b+6=103b+16=103b=−6b=−25b + 10 - 2b + 6 = 10 \\ 3b + 16 = 10 \\ 3b = -6 \\ b = -
25b+10−2b+6=103b+16=103b=−6b=−2
8. Solving the equation:
(x+1)2−4(x−2)=0(x+1)^2 - 4(x-2) = 0(x+1)2−4(x−2)=0
Expand and simplify:
(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0(x+1)^2 - 4x + 8 = 0 \\ x^2 + 2x + 1 - 4x + 8 =
0 \\ x^2 - 2x + 9 = 0(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0
Solve using the quadratic formula x=−b±b2−4ac2ax = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}x=2a−b±b2−4ac:
x=2±4−362x=2±−322x=1±i8x = \{2 \pm \sqrt{4 - 36}}{2} \\ x = \{2 \pm \sqrt{-32}}{2} \\ x
= 1 \pm i\sqrt{8}x=22±4−36x=22±−32x=1±i8
So, x=1+22ix = 1 + 2\sqrt{2}ix=1+22i and x=1−22ix = 1 - 2\sqrt{2}ix=1−22i.
9. Solving the equation:
3(y−1)2+2(y+3)=03(y-1)^2 + 2(y+3) = 03(y−1)2+2(y+3)=0
Expand and combine like terms:
3(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=03(y^2 - 2y + 1) + 2y + 6 = 0 \\ 3y^2 - 6y
+ 3 + 2y + 6 = 0 \\ 3y^2 - 4y + 9 = 03(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=0
Solve using the quadratic formula y=−b±b2−4ac2ay = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}y=2a−b±b2−4ac:
y=4±16−1086y=4±−926y=4±2i236y=2±i233y = \{4 \pm \sqrt{16 - 108}}{6} \\ y = \{4 \pm \
sqrt{-92}}{6} \\ y = \{4 \pm 2i\sqrt{23}}{6} \\ y = \{2 \pm i\sqrt{23}}{3}y=64±16−108
y=64±−92y=64±2i23y=32±i23
So, y=2+i233y = \{2 + i\sqrt{23}}{3}y=32+i23 and y=2−i233y = \{2 - i\sqrt{23}}
{3}y=32−i23.
10. Solving the equation:
(z−2)2−5(z+1)=0(z-2)^2 - 5(z+1) = 0(z−2)2−5(z+1)=0
Expand and simplify:
z2−4z+4−5z−5=0z2−9z−1=0z^2 - 4z + 4 - 5z - 5 = 0 \\ z^2 - 9z - 1 =
0z2−4z+4−5z−5=0z2−9z−1=0
Solve using the quadratic formula z=−b±b2−4ac2az = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}z=2a−b±b2−4ac:
z=9±81+42z=9±852z = \{9 \pm \sqrt{81 + 4}}{2} \\ z = \{9 \pm \sqrt{85}}{2}z=29±81+4
z=29±85
So, z=9+852z = \{9 + \sqrt{85}}{2}z=29+85 and z=9−852z = \{9 - \sqrt{85}}{2}z=29−85.
Advanced Algebraic Expressions
61. Simplify:
3x2−9x3x\frac{3x^2 - 9x}{3x}3x3x2−9x
Factor the numerator:
3x(x−3)3x=x−3\frac{3x(x - 3)}{3x} = x - 33x3x(x−3)=x−3
62. Simplify:
4y2−16y4y\frac{4y^2 - 16y}{4y}4y4y2−16y
Factor the numerator:
4y(y−4)4y=y−4\frac{4y(y - 4)}{4y} = y - 44y4y(y−4)=y−4
63. Simplify:
2z2+8z2z\frac{2z^2 + 8z}{2z}2z2z2+8z
Factor the numerator:
2z(z+4)2z=z+4\frac{2z(z + 4)}{2z} = z + 42z2z(z+4)=z+4
64. Expand:
(2x−3)2(2x - 3)^2(2x−3)2
Use the binomial expansion formula (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab +
b^2(a−b)2=a2−2ab+b2:
(2x−3)2=4x2−12x+9(2x - 3)^2 = 4x^2 - 12x + 9(2x−3)2=4x2−12x+9
65. Expand:
(y+4)2(y + 4)^2(y+4)2
Use the binomial expansion formula (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab +
b^2(a+b)2=a2+2ab+b2:
(y+4)2=y2+8y+16(y + 4)^2 = y^2 + 8y + 16(y+4)2=y2+8y+16
66. Expand:
(3a−2b)(3a+2b)(3a - 2b)(3a + 2b)(3a−2b)(3a+2b)
Use the difference of squares formula (a−b)(a+b)=a2−b2(a - b)(a + b) = a^2 - b^2(a−b)
(a+b)=a2−b2:
(3a−2b)(3a+2b)=9a2−4b2(3a - 2b)(3a + 2b) = 9a^2 - 4b^2(3a−2b)(3a+2b)=9a2−4b2
67. Factor:
x2−9x^2 - 9x2−9
This is a difference of squares:
5. Solving the equation:
3z−14+z+23=5\{3z-1}{4} + \{z+2}{3} = 543z−1+3z+2=5
Find a common denominator (12):
3(3z−1)12+4(z+2)12=59z−3+4z+812=513z+512=513z+5=6013z=55z=5513≈4.23\{3(3z-1)}
{12} + \{4(z+2)}{12} = 5 \\ \{9z - 3 + 4z + 8}{12} = 5 \\ \{13z + 5}{12} = 5 \\ 13z + 5 =
60 \\ 13z = 55 \\ z = \{55}{13} \approx 4.23123(3z−1)+124(z+2)=5129z−3+4z+8=51213z+5
=513z+5=6013z=55z=1355≈4.23
6. Solving the equation:
2(a−3)+3(a+4)=52(a-3) + 3(a+4) = 52(a−3)+3(a+4)=5
Expand and combine like terms:
2a−6+3a+12=55a+6=55a=−1a=−152a - 6 + 3a + 12 = 5 \\ 5a + 6 = 5 \\ 5a = -1 \\ a = -\{1}
{5}2a−6+3a+12=55a+6=55a=−1a=−51
7. Solving the equation:
5(b+2)−2(b−3)=105(b+2) - 2(b-3) = 105(b+2)−2(b−3)=10
Expand and combine like terms:
5b+10−2b+6=103b+16=103b=−6b=−25b + 10 - 2b + 6 = 10 \\ 3b + 16 = 10 \\ 3b = -6 \\ b = -
25b+10−2b+6=103b+16=103b=−6b=−2
8. Solving the equation:
(x+1)2−4(x−2)=0(x+1)^2 - 4(x-2) = 0(x+1)2−4(x−2)=0
Expand and simplify:
(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0(x+1)^2 - 4x + 8 = 0 \\ x^2 + 2x + 1 - 4x + 8 =
0 \\ x^2 - 2x + 9 = 0(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0
Solve using the quadratic formula x=−b±b2−4ac2ax = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}x=2a−b±b2−4ac:
x=2±4−362x=2±−322x=1±i8x = \{2 \pm \sqrt{4 - 36}}{2} \\ x = \{2 \pm \sqrt{-32}}{2} \\ x
= 1 \pm i\sqrt{8}x=22±4−36x=22±−32x=1±i8
So, x=1+22ix = 1 + 2\sqrt{2}ix=1+22i and x=1−22ix = 1 - 2\sqrt{2}ix=1−22i.
9. Solving the equation:
3(y−1)2+2(y+3)=03(y-1)^2 + 2(y+3) = 03(y−1)2+2(y+3)=0
Expand and combine like terms:
3(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=03(y^2 - 2y + 1) + 2y + 6 = 0 \\ 3y^2 - 6y
+ 3 + 2y + 6 = 0 \\ 3y^2 - 4y + 9 = 03(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=0
Solve using the quadratic formula y=−b±b2−4ac2ay = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}y=2a−b±b2−4ac:
y=4±16−1086y=4±−926y=4±2i236y=2±i233y = \{4 \pm \sqrt{16 - 108}}{6} \\ y = \{4 \pm \
sqrt{-92}}{6} \\ y = \{4 \pm 2i\sqrt{23}}{6} \\ y = \{2 \pm i\sqrt{23}}{3}y=64±16−108
y=64±−92y=64±2i23y=32±i23
So, y=2+i233y = \{2 + i\sqrt{23}}{3}y=32+i23 and y=2−i233y = \{2 - i\sqrt{23}}
{3}y=32−i23.
10. Solving the equation:
(z−2)2−5(z+1)=0(z-2)^2 - 5(z+1) = 0(z−2)2−5(z+1)=0
Expand and simplify:
z2−4z+4−5z−5=0z2−9z−1=0z^2 - 4z + 4 - 5z - 5 = 0 \\ z^2 - 9z - 1 =
0z2−4z+4−5z−5=0z2−9z−1=0
Solve using the quadratic formula z=−b±b2−4ac2az = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}z=2a−b±b2−4ac:
z=9±81+42z=9±852z = \{9 \pm \sqrt{81 + 4}}{2} \\ z = \{9 \pm \sqrt{85}}{2}z=29±81+4
z=29±85
So, z=9+852z = \{9 + \sqrt{85}}{2}z=29+85 and z=9−852z = \{9 - \sqrt{85}}{2}z=29−85.
Advanced Algebraic Expressions
61. Simplify:
3x2−9x3x\frac{3x^2 - 9x}{3x}3x3x2−9x
Factor the numerator:
3x(x−3)3x=x−3\frac{3x(x - 3)}{3x} = x - 33x3x(x−3)=x−3
62. Simplify:
4y2−16y4y\frac{4y^2 - 16y}{4y}4y4y2−16y
Factor the numerator:
4y(y−4)4y=y−4\frac{4y(y - 4)}{4y} = y - 44y4y(y−4)=y−4
63. Simplify:
2z2+8z2z\frac{2z^2 + 8z}{2z}2z2z2+8z
Factor the numerator:
2z(z+4)2z=z+4\frac{2z(z + 4)}{2z} = z + 42z2z(z+4)=z+4
64. Expand:
(2x−3)2(2x - 3)^2(2x−3)2
Use the binomial expansion formula (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab +
b^2(a−b)2=a2−2ab+b2:
(2x−3)2=4x2−12x+9(2x - 3)^2 = 4x^2 - 12x + 9(2x−3)2=4x2−12x+9
65. Expand:
(y+4)2(y + 4)^2(y+4)2
Use the binomial expansion formula (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab +
b^2(a+b)2=a2+2ab+b2:
(y+4)2=y2+8y+16(y + 4)^2 = y^2 + 8y + 16(y+4)2=y2+8y+16
66. Expand:
(3a−2b)(3a+2b)(3a - 2b)(3a + 2b)(3a−2b)(3a+2b)
Use the difference of squares formula (a−b)(a+b)=a2−b2(a - b)(a + b) = a^2 - b^2(a−b)
(a+b)=a2−b2:
(3a−2b)(3a+2b)=9a2−4b2(3a - 2b)(3a + 2b) = 9a^2 - 4b^2(3a−2b)(3a+2b)=9a2−4b2
67. Factor:
x2−9x^2 - 9x2−9
This is a difference of squares:
5. Solving the equation:
3z−14+z+23=5\{3z-1}{4} + \{z+2}{3} = 543z−1+3z+2=5
Find a common denominator (12):
3(3z−1)12+4(z+2)12=59z−3+4z+812=513z+512=513z+5=6013z=55z=5513≈4.23\{3(3z-1)}
{12} + \{4(z+2)}{12} = 5 \\ \{9z - 3 + 4z + 8}{12} = 5 \\ \{13z + 5}{12} = 5 \\ 13z + 5 =
60 \\ 13z = 55 \\ z = \{55}{13} \approx 4.23123(3z−1)+124(z+2)=5129z−3+4z+8=51213z+5
=513z+5=6013z=55z=1355≈4.23
6. Solving the equation:
2(a−3)+3(a+4)=52(a-3) + 3(a+4) = 52(a−3)+3(a+4)=5
Expand and combine like terms:
2a−6+3a+12=55a+6=55a=−1a=−152a - 6 + 3a + 12 = 5 \\ 5a + 6 = 5 \\ 5a = -1 \\ a = -\{1}
{5}2a−6+3a+12=55a+6=55a=−1a=−51
7. Solving the equation:
5(b+2)−2(b−3)=105(b+2) - 2(b-3) = 105(b+2)−2(b−3)=10
Expand and combine like terms:
5b+10−2b+6=103b+16=103b=−6b=−25b + 10 - 2b + 6 = 10 \\ 3b + 16 = 10 \\ 3b = -6 \\ b = -
25b+10−2b+6=103b+16=103b=−6b=−2
8. Solving the equation:
(x+1)2−4(x−2)=0(x+1)^2 - 4(x-2) = 0(x+1)2−4(x−2)=0
Expand and simplify:
(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0(x+1)^2 - 4x + 8 = 0 \\ x^2 + 2x + 1 - 4x + 8 =
0 \\ x^2 - 2x + 9 = 0(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0
Solve using the quadratic formula x=−b±b2−4ac2ax = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}x=2a−b±b2−4ac:
x=2±4−362x=2±−322x=1±i8x = \{2 \pm \sqrt{4 - 36}}{2} \\ x = \{2 \pm \sqrt{-32}}{2} \\ x
= 1 \pm i\sqrt{8}x=22±4−36x=22±−32x=1±i8
So, x=1+22ix = 1 + 2\sqrt{2}ix=1+22i and x=1−22ix = 1 - 2\sqrt{2}ix=1−22i.
9. Solving the equation:
3(y−1)2+2(y+3)=03(y-1)^2 + 2(y+3) = 03(y−1)2+2(y+3)=0
Expand and combine like terms:
3(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=03(y^2 - 2y + 1) + 2y + 6 = 0 \\ 3y^2 - 6y
+ 3 + 2y + 6 = 0 \\ 3y^2 - 4y + 9 = 03(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=0
Solve using the quadratic formula y=−b±b2−4ac2ay = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}y=2a−b±b2−4ac:
y=4±16−1086y=4±−926y=4±2i236y=2±i233y = \{4 \pm \sqrt{16 - 108}}{6} \\ y = \{4 \pm \
sqrt{-92}}{6} \\ y = \{4 \pm 2i\sqrt{23}}{6} \\ y = \{2 \pm i\sqrt{23}}{3}y=64±16−108
y=64±−92y=64±2i23y=32±i23
So, y=2+i233y = \{2 + i\sqrt{23}}{3}y=32+i23 and y=2−i233y = \{2 - i\sqrt{23}}
{3}y=32−i23.
10. Solving the equation:
(z−2)2−5(z+1)=0(z-2)^2 - 5(z+1) = 0(z−2)2−5(z+1)=0
Expand and simplify:
z2−4z+4−5z−5=0z2−9z−1=0z^2 - 4z + 4 - 5z - 5 = 0 \\ z^2 - 9z - 1 =
0z2−4z+4−5z−5=0z2−9z−1=0
Solve using the quadratic formula z=−b±b2−4ac2az = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}z=2a−b±b2−4ac:
z=9±81+42z=9±852z = \{9 \pm \sqrt{81 + 4}}{2} \\ z = \{9 \pm \sqrt{85}}{2}z=29±81+4
z=29±85
So, z=9+852z = \{9 + \sqrt{85}}{2}z=29+85 and z=9−852z = \{9 - \sqrt{85}}{2}z=29−85.
Advanced Algebraic Expressions
61. Simplify:
3x2−9x3x\frac{3x^2 - 9x}{3x}3x3x2−9x
Factor the numerator:
3x(x−3)3x=x−3\frac{3x(x - 3)}{3x} = x - 33x3x(x−3)=x−3
62. Simplify:
4y2−16y4y\frac{4y^2 - 16y}{4y}4y4y2−16y
Factor the numerator:
4y(y−4)4y=y−4\frac{4y(y - 4)}{4y} = y - 44y4y(y−4)=y−4
63. Simplify:
2z2+8z2z\frac{2z^2 + 8z}{2z}2z2z2+8z
Factor the numerator:
2z(z+4)2z=z+4\frac{2z(z + 4)}{2z} = z + 42z2z(z+4)=z+4
64. Expand:
(2x−3)2(2x - 3)^2(2x−3)2
Use the binomial expansion formula (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab +
b^2(a−b)2=a2−2ab+b2:
(2x−3)2=4x2−12x+9(2x - 3)^2 = 4x^2 - 12x + 9(2x−3)2=4x2−12x+9
65. Expand:
(y+4)2(y + 4)^2(y+4)2
Use the binomial expansion formula (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab +
b^2(a+b)2=a2+2ab+b2:
(y+4)2=y2+8y+16(y + 4)^2 = y^2 + 8y + 16(y+4)2=y2+8y+16
66. Expand:
(3a−2b)(3a+2b)(3a - 2b)(3a + 2b)(3a−2b)(3a+2b)
Use the difference of squares formula (a−b)(a+b)=a2−b2(a - b)(a + b) = a^2 - b^2(a−b)
(a+b)=a2−b2:
(3a−2b)(3a+2b)=9a2−4b2(3a - 2b)(3a + 2b) = 9a^2 - 4b^2(3a−2b)(3a+2b)=9a2−4b2
67. Factor:
x2−9x^2 - 9x2−9
This is a difference of squares:
5. Solving the equation:
3z−14+z+23=5\{3z-1}{4} + \{z+2}{3} = 543z−1+3z+2=5
Find a common denominator (12):
3(3z−1)12+4(z+2)12=59z−3+4z+812=513z+512=513z+5=6013z=55z=5513≈4.23\{3(3z-1)}
{12} + \{4(z+2)}{12} = 5 \\ \{9z - 3 + 4z + 8}{12} = 5 \\ \{13z + 5}{12} = 5 \\ 13z + 5 =
60 \\ 13z = 55 \\ z = \{55}{13} \approx 4.23123(3z−1)+124(z+2)=5129z−3+4z+8=51213z+5
=513z+5=6013z=55z=1355≈4.23
6. Solving the equation:
2(a−3)+3(a+4)=52(a-3) + 3(a+4) = 52(a−3)+3(a+4)=5
Expand and combine like terms:
2a−6+3a+12=55a+6=55a=−1a=−152a - 6 + 3a + 12 = 5 \\ 5a + 6 = 5 \\ 5a = -1 \\ a = -\{1}
{5}2a−6+3a+12=55a+6=55a=−1a=−51
7. Solving the equation:
5(b+2)−2(b−3)=105(b+2) - 2(b-3) = 105(b+2)−2(b−3)=10
Expand and combine like terms:
5b+10−2b+6=103b+16=103b=−6b=−25b + 10 - 2b + 6 = 10 \\ 3b + 16 = 10 \\ 3b = -6 \\ b = -
25b+10−2b+6=103b+16=103b=−6b=−2
8. Solving the equation:
(x+1)2−4(x−2)=0(x+1)^2 - 4(x-2) = 0(x+1)2−4(x−2)=0
Expand and simplify:
(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0(x+1)^2 - 4x + 8 = 0 \\ x^2 + 2x + 1 - 4x + 8 =
0 \\ x^2 - 2x + 9 = 0(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0
Solve using the quadratic formula x=−b±b2−4ac2ax = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}x=2a−b±b2−4ac:
x=2±4−362x=2±−322x=1±i8x = \{2 \pm \sqrt{4 - 36}}{2} \\ x = \{2 \pm \sqrt{-32}}{2} \\ x
= 1 \pm i\sqrt{8}x=22±4−36x=22±−32x=1±i8
So, x=1+22ix = 1 + 2\sqrt{2}ix=1+22i and x=1−22ix = 1 - 2\sqrt{2}ix=1−22i.
9. Solving the equation:
3(y−1)2+2(y+3)=03(y-1)^2 + 2(y+3) = 03(y−1)2+2(y+3)=0
Expand and combine like terms:
3(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=03(y^2 - 2y + 1) + 2y + 6 = 0 \\ 3y^2 - 6y
+ 3 + 2y + 6 = 0 \\ 3y^2 - 4y + 9 = 03(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=0
Solve using the quadratic formula y=−b±b2−4ac2ay = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}y=2a−b±b2−4ac:
y=4±16−1086y=4±−926y=4±2i236y=2±i233y = \{4 \pm \sqrt{16 - 108}}{6} \\ y = \{4 \pm \
sqrt{-92}}{6} \\ y = \{4 \pm 2i\sqrt{23}}{6} \\ y = \{2 \pm i\sqrt{23}}{3}y=64±16−108
y=64±−92y=64±2i23y=32±i23
So, y=2+i233y = \{2 + i\sqrt{23}}{3}y=32+i23 and y=2−i233y = \{2 - i\sqrt{23}}
{3}y=32−i23.
10. Solving the equation:
(z−2)2−5(z+1)=0(z-2)^2 - 5(z+1) = 0(z−2)2−5(z+1)=0
Expand and simplify:
z2−4z+4−5z−5=0z2−9z−1=0z^2 - 4z + 4 - 5z - 5 = 0 \\ z^2 - 9z - 1 =
0z2−4z+4−5z−5=0z2−9z−1=0
Solve using the quadratic formula z=−b±b2−4ac2az = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}z=2a−b±b2−4ac:
z=9±81+42z=9±852z = \{9 \pm \sqrt{81 + 4}}{2} \\ z = \{9 \pm \sqrt{85}}{2}z=29±81+4
z=29±85
So, z=9+852z = \{9 + \sqrt{85}}{2}z=29+85 and z=9−852z = \{9 - \sqrt{85}}{2}z=29−85.
Advanced Algebraic Expressions
61. Simplify:
3x2−9x3x\frac{3x^2 - 9x}{3x}3x3x2−9x
Factor the numerator:
3x(x−3)3x=x−3\frac{3x(x - 3)}{3x} = x - 33x3x(x−3)=x−3
62. Simplify:
4y2−16y4y\frac{4y^2 - 16y}{4y}4y4y2−16y
Factor the numerator:
4y(y−4)4y=y−4\frac{4y(y - 4)}{4y} = y - 44y4y(y−4)=y−4
63. Simplify:
2z2+8z2z\frac{2z^2 + 8z}{2z}2z2z2+8z
Factor the numerator:
2z(z+4)2z=z+4\frac{2z(z + 4)}{2z} = z + 42z2z(z+4)=z+4
64. Expand:
(2x−3)2(2x - 3)^2(2x−3)2
Use the binomial expansion formula (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab +
b^2(a−b)2=a2−2ab+b2:
(2x−3)2=4x2−12x+9(2x - 3)^2 = 4x^2 - 12x + 9(2x−3)2=4x2−12x+9
65. Expand:
(y+4)2(y + 4)^2(y+4)2
Use the binomial expansion formula (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab +
b^2(a+b)2=a2+2ab+b2:
(y+4)2=y2+8y+16(y + 4)^2 = y^2 + 8y + 16(y+4)2=y2+8y+16
66. Expand:
(3a−2b)(3a+2b)(3a - 2b)(3a + 2b)(3a−2b)(3a+2b)
Use the difference of squares formula (a−b)(a+b)=a2−b2(a - b)(a + b) = a^2 - b^2(a−b)
(a+b)=a2−b2:
(3a−2b)(3a+2b)=9a2−4b2(3a - 2b)(3a + 2b) = 9a^2 - 4b^2(3a−2b)(3a+2b)=9a2−4b2
67. Factor:
x2−9x^2 - 9x2−9
This is a difference of squares:
5. Solving the equation:
3z−14+z+23=5\{3z-1}{4} + \{z+2}{3} = 543z−1+3z+2=5
Find a common denominator (12):
3(3z−1)12+4(z+2)12=59z−3+4z+812=513z+512=513z+5=6013z=55z=5513≈4.23\{3(3z-1)}
{12} + \{4(z+2)}{12} = 5 \\ \{9z - 3 + 4z + 8}{12} = 5 \\ \{13z + 5}{12} = 5 \\ 13z + 5 =
60 \\ 13z = 55 \\ z = \{55}{13} \approx 4.23123(3z−1)+124(z+2)=5129z−3+4z+8=51213z+5
=513z+5=6013z=55z=1355≈4.23
6. Solving the equation:
2(a−3)+3(a+4)=52(a-3) + 3(a+4) = 52(a−3)+3(a+4)=5
Expand and combine like terms:
2a−6+3a+12=55a+6=55a=−1a=−152a - 6 + 3a + 12 = 5 \\ 5a + 6 = 5 \\ 5a = -1 \\ a = -\{1}
{5}2a−6+3a+12=55a+6=55a=−1a=−51
7. Solving the equation:
5(b+2)−2(b−3)=105(b+2) - 2(b-3) = 105(b+2)−2(b−3)=10
Expand and combine like terms:
5b+10−2b+6=103b+16=103b=−6b=−25b + 10 - 2b + 6 = 10 \\ 3b + 16 = 10 \\ 3b = -6 \\ b = -
25b+10−2b+6=103b+16=103b=−6b=−2
8. Solving the equation:
(x+1)2−4(x−2)=0(x+1)^2 - 4(x-2) = 0(x+1)2−4(x−2)=0
Expand and simplify:
(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0(x+1)^2 - 4x + 8 = 0 \\ x^2 + 2x + 1 - 4x + 8 =
0 \\ x^2 - 2x + 9 = 0(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0
Solve using the quadratic formula x=−b±b2−4ac2ax = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}x=2a−b±b2−4ac:
x=2±4−362x=2±−322x=1±i8x = \{2 \pm \sqrt{4 - 36}}{2} \\ x = \{2 \pm \sqrt{-32}}{2} \\ x
= 1 \pm i\sqrt{8}x=22±4−36x=22±−32x=1±i8
So, x=1+22ix = 1 + 2\sqrt{2}ix=1+22i and x=1−22ix = 1 - 2\sqrt{2}ix=1−22i.
9. Solving the equation:
3(y−1)2+2(y+3)=03(y-1)^2 + 2(y+3) = 03(y−1)2+2(y+3)=0
Expand and combine like terms:
3(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=03(y^2 - 2y + 1) + 2y + 6 = 0 \\ 3y^2 - 6y
+ 3 + 2y + 6 = 0 \\ 3y^2 - 4y + 9 = 03(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=0
Solve using the quadratic formula y=−b±b2−4ac2ay = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}y=2a−b±b2−4ac:
y=4±16−1086y=4±−926y=4±2i236y=2±i233y = \{4 \pm \sqrt{16 - 108}}{6} \\ y = \{4 \pm \
sqrt{-92}}{6} \\ y = \{4 \pm 2i\sqrt{23}}{6} \\ y = \{2 \pm i\sqrt{23}}{3}y=64±16−108
y=64±−92y=64±2i23y=32±i23
So, y=2+i233y = \{2 + i\sqrt{23}}{3}y=32+i23 and y=2−i233y = \{2 - i\sqrt{23}}
{3}y=32−i23.
10. Solving the equation:
(z−2)2−5(z+1)=0(z-2)^2 - 5(z+1) = 0(z−2)2−5(z+1)=0
Expand and simplify:
z2−4z+4−5z−5=0z2−9z−1=0z^2 - 4z + 4 - 5z - 5 = 0 \\ z^2 - 9z - 1 =
0z2−4z+4−5z−5=0z2−9z−1=0
Solve using the quadratic formula z=−b±b2−4ac2az = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}z=2a−b±b2−4ac:
z=9±81+42z=9±852z = \{9 \pm \sqrt{81 + 4}}{2} \\ z = \{9 \pm \sqrt{85}}{2}z=29±81+4
z=29±85
So, z=9+852z = \{9 + \sqrt{85}}{2}z=29+85 and z=9−852z = \{9 - \sqrt{85}}{2}z=29−85.
Advanced Algebraic Expressions
61. Simplify:
3x2−9x3x\frac{3x^2 - 9x}{3x}3x3x2−9x
Factor the numerator:
3x(x−3)3x=x−3\frac{3x(x - 3)}{3x} = x - 33x3x(x−3)=x−3
62. Simplify:
4y2−16y4y\frac{4y^2 - 16y}{4y}4y4y2−16y
Factor the numerator:
4y(y−4)4y=y−4\frac{4y(y - 4)}{4y} = y - 44y4y(y−4)=y−4
63. Simplify:
2z2+8z2z\frac{2z^2 + 8z}{2z}2z2z2+8z
Factor the numerator:
2z(z+4)2z=z+4\frac{2z(z + 4)}{2z} = z + 42z2z(z+4)=z+4
64. Expand:
(2x−3)2(2x - 3)^2(2x−3)2
Use the binomial expansion formula (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab +
b^2(a−b)2=a2−2ab+b2:
(2x−3)2=4x2−12x+9(2x - 3)^2 = 4x^2 - 12x + 9(2x−3)2=4x2−12x+9
65. Expand:
(y+4)2(y + 4)^2(y+4)2
Use the binomial expansion formula (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab +
b^2(a+b)2=a2+2ab+b2:
(y+4)2=y2+8y+16(y + 4)^2 = y^2 + 8y + 16(y+4)2=y2+8y+16
66. Expand:
(3a−2b)(3a+2b)(3a - 2b)(3a + 2b)(3a−2b)(3a+2b)
Use the difference of squares formula (a−b)(a+b)=a2−b2(a - b)(a + b) = a^2 - b^2(a−b)
(a+b)=a2−b2:
(3a−2b)(3a+2b)=9a2−4b2(3a - 2b)(3a + 2b) = 9a^2 - 4b^2(3a−2b)(3a+2b)=9a2−4b2
67. Factor:
x2−9x^2 - 9x2−9
This is a difference of squares:
5. Solving the equation:
3z−14+z+23=5\{3z-1}{4} + \{z+2}{3} = 543z−1+3z+2=5
Find a common denominator (12):
3(3z−1)12+4(z+2)12=59z−3+4z+812=513z+512=513z+5=6013z=55z=5513≈4.23\{3(3z-1)}
{12} + \{4(z+2)}{12} = 5 \\ \{9z - 3 + 4z + 8}{12} = 5 \\ \{13z + 5}{12} = 5 \\ 13z + 5 =
60 \\ 13z = 55 \\ z = \{55}{13} \approx 4.23123(3z−1)+124(z+2)=5129z−3+4z+8=51213z+5
=513z+5=6013z=55z=1355≈4.23
6. Solving the equation:
2(a−3)+3(a+4)=52(a-3) + 3(a+4) = 52(a−3)+3(a+4)=5
Expand and combine like terms:
2a−6+3a+12=55a+6=55a=−1a=−152a - 6 + 3a + 12 = 5 \\ 5a + 6 = 5 \\ 5a = -1 \\ a = -\{1}
{5}2a−6+3a+12=55a+6=55a=−1a=−51
7. Solving the equation:
5(b+2)−2(b−3)=105(b+2) - 2(b-3) = 105(b+2)−2(b−3)=10
Expand and combine like terms:
5b+10−2b+6=103b+16=103b=−6b=−25b + 10 - 2b + 6 = 10 \\ 3b + 16 = 10 \\ 3b = -6 \\ b = -
25b+10−2b+6=103b+16=103b=−6b=−2
8. Solving the equation:
(x+1)2−4(x−2)=0(x+1)^2 - 4(x-2) = 0(x+1)2−4(x−2)=0
Expand and simplify:
(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0(x+1)^2 - 4x + 8 = 0 \\ x^2 + 2x + 1 - 4x + 8 =
0 \\ x^2 - 2x + 9 = 0(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0
Solve using the quadratic formula x=−b±b2−4ac2ax = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}x=2a−b±b2−4ac:
x=2±4−362x=2±−322x=1±i8x = \{2 \pm \sqrt{4 - 36}}{2} \\ x = \{2 \pm \sqrt{-32}}{2} \\ x
= 1 \pm i\sqrt{8}x=22±4−36x=22±−32x=1±i8
So, x=1+22ix = 1 + 2\sqrt{2}ix=1+22i and x=1−22ix = 1 - 2\sqrt{2}ix=1−22i.
9. Solving the equation:
3(y−1)2+2(y+3)=03(y-1)^2 + 2(y+3) = 03(y−1)2+2(y+3)=0
Expand and combine like terms:
3(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=03(y^2 - 2y + 1) + 2y + 6 = 0 \\ 3y^2 - 6y
+ 3 + 2y + 6 = 0 \\ 3y^2 - 4y + 9 = 03(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=0
Solve using the quadratic formula y=−b±b2−4ac2ay = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}y=2a−b±b2−4ac:
y=4±16−1086y=4±−926y=4±2i236y=2±i233y = \{4 \pm \sqrt{16 - 108}}{6} \\ y = \{4 \pm \
sqrt{-92}}{6} \\ y = \{4 \pm 2i\sqrt{23}}{6} \\ y = \{2 \pm i\sqrt{23}}{3}y=64±16−108
y=64±−92y=64±2i23y=32±i23
So, y=2+i233y = \{2 + i\sqrt{23}}{3}y=32+i23 and y=2−i233y = \{2 - i\sqrt{23}}
{3}y=32−i23.
10. Solving the equation:
(z−2)2−5(z+1)=0(z-2)^2 - 5(z+1) = 0(z−2)2−5(z+1)=0
Expand and simplify:
z2−4z+4−5z−5=0z2−9z−1=0z^2 - 4z + 4 - 5z - 5 = 0 \\ z^2 - 9z - 1 =
0z2−4z+4−5z−5=0z2−9z−1=0
Solve using the quadratic formula z=−b±b2−4ac2az = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}z=2a−b±b2−4ac:
z=9±81+42z=9±852z = \{9 \pm \sqrt{81 + 4}}{2} \\ z = \{9 \pm \sqrt{85}}{2}z=29±81+4
z=29±85
So, z=9+852z = \{9 + \sqrt{85}}{2}z=29+85 and z=9−852z = \{9 - \sqrt{85}}{2}z=29−85.
Advanced Algebraic Expressions
61. Simplify:
3x2−9x3x\frac{3x^2 - 9x}{3x}3x3x2−9x
Factor the numerator:
3x(x−3)3x=x−3\frac{3x(x - 3)}{3x} = x - 33x3x(x−3)=x−3
62. Simplify:
4y2−16y4y\frac{4y^2 - 16y}{4y}4y4y2−16y
Factor the numerator:
4y(y−4)4y=y−4\frac{4y(y - 4)}{4y} = y - 44y4y(y−4)=y−4
63. Simplify:
2z2+8z2z\frac{2z^2 + 8z}{2z}2z2z2+8z
Factor the numerator:
2z(z+4)2z=z+4\frac{2z(z + 4)}{2z} = z + 42z2z(z+4)=z+4
64. Expand:
(2x−3)2(2x - 3)^2(2x−3)2
Use the binomial expansion formula (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab +
b^2(a−b)2=a2−2ab+b2:
(2x−3)2=4x2−12x+9(2x - 3)^2 = 4x^2 - 12x + 9(2x−3)2=4x2−12x+9
65. Expand:
(y+4)2(y + 4)^2(y+4)2
Use the binomial expansion formula (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab +
b^2(a+b)2=a2+2ab+b2:
(y+4)2=y2+8y+16(y + 4)^2 = y^2 + 8y + 16(y+4)2=y2+8y+16
66. Expand:
(3a−2b)(3a+2b)(3a - 2b)(3a + 2b)(3a−2b)(3a+2b)
Use the difference of squares formula (a−b)(a+b)=a2−b2(a - b)(a + b) = a^2 - b^2(a−b)
(a+b)=a2−b2:
(3a−2b)(3a+2b)=9a2−4b2(3a - 2b)(3a + 2b) = 9a^2 - 4b^2(3a−2b)(3a+2b)=9a2−4b2
67. Factor:
x2−9x^2 - 9x2−9
This is a difference of squares:
5. Solving the equation:
3z−14+z+23=5\{3z-1}{4} + \{z+2}{3} = 543z−1+3z+2=5
Find a common denominator (12):
3(3z−1)12+4(z+2)12=59z−3+4z+812=513z+512=513z+5=6013z=55z=5513≈4.23\{3(3z-1)}
{12} + \{4(z+2)}{12} = 5 \\ \{9z - 3 + 4z + 8}{12} = 5 \\ \{13z + 5}{12} = 5 \\ 13z + 5 =
60 \\ 13z = 55 \\ z = \{55}{13} \approx 4.23123(3z−1)+124(z+2)=5129z−3+4z+8=51213z+5
=513z+5=6013z=55z=1355≈4.23
6. Solving the equation:
2(a−3)+3(a+4)=52(a-3) + 3(a+4) = 52(a−3)+3(a+4)=5
Expand and combine like terms:
2a−6+3a+12=55a+6=55a=−1a=−152a - 6 + 3a + 12 = 5 \\ 5a + 6 = 5 \\ 5a = -1 \\ a = -\{1}
{5}2a−6+3a+12=55a+6=55a=−1a=−51
7. Solving the equation:
5(b+2)−2(b−3)=105(b+2) - 2(b-3) = 105(b+2)−2(b−3)=10
Expand and combine like terms:
5b+10−2b+6=103b+16=103b=−6b=−25b + 10 - 2b + 6 = 10 \\ 3b + 16 = 10 \\ 3b = -6 \\ b = -
25b+10−2b+6=103b+16=103b=−6b=−2
8. Solving the equation:
(x+1)2−4(x−2)=0(x+1)^2 - 4(x-2) = 0(x+1)2−4(x−2)=0
Expand and simplify:
(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0(x+1)^2 - 4x + 8 = 0 \\ x^2 + 2x + 1 - 4x + 8 =
0 \\ x^2 - 2x + 9 = 0(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0
Solve using the quadratic formula x=−b±b2−4ac2ax = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}x=2a−b±b2−4ac:
x=2±4−362x=2±−322x=1±i8x = \{2 \pm \sqrt{4 - 36}}{2} \\ x = \{2 \pm \sqrt{-32}}{2} \\ x
= 1 \pm i\sqrt{8}x=22±4−36x=22±−32x=1±i8
So, x=1+22ix = 1 + 2\sqrt{2}ix=1+22i and x=1−22ix = 1 - 2\sqrt{2}ix=1−22i.
9. Solving the equation:
3(y−1)2+2(y+3)=03(y-1)^2 + 2(y+3) = 03(y−1)2+2(y+3)=0
Expand and combine like terms:
3(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=03(y^2 - 2y + 1) + 2y + 6 = 0 \\ 3y^2 - 6y
+ 3 + 2y + 6 = 0 \\ 3y^2 - 4y + 9 = 03(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=0
Solve using the quadratic formula y=−b±b2−4ac2ay = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}y=2a−b±b2−4ac:
y=4±16−1086y=4±−926y=4±2i236y=2±i233y = \{4 \pm \sqrt{16 - 108}}{6} \\ y = \{4 \pm \
sqrt{-92}}{6} \\ y = \{4 \pm 2i\sqrt{23}}{6} \\ y = \{2 \pm i\sqrt{23}}{3}y=64±16−108
y=64±−92y=64±2i23y=32±i23
So, y=2+i233y = \{2 + i\sqrt{23}}{3}y=32+i23 and y=2−i233y = \{2 - i\sqrt{23}}
{3}y=32−i23.
10. Solving the equation:
(z−2)2−5(z+1)=0(z-2)^2 - 5(z+1) = 0(z−2)2−5(z+1)=0
Expand and simplify:
z2−4z+4−5z−5=0z2−9z−1=0z^2 - 4z + 4 - 5z - 5 = 0 \\ z^2 - 9z - 1 =
0z2−4z+4−5z−5=0z2−9z−1=0
Solve using the quadratic formula z=−b±b2−4ac2az = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}z=2a−b±b2−4ac:
z=9±81+42z=9±852z = \{9 \pm \sqrt{81 + 4}}{2} \\ z = \{9 \pm \sqrt{85}}{2}z=29±81+4
z=29±85
So, z=9+852z = \{9 + \sqrt{85}}{2}z=29+85 and z=9−852z = \{9 - \sqrt{85}}{2}z=29−85.
Advanced Algebraic Expressions
61. Simplify:
3x2−9x3x\frac{3x^2 - 9x}{3x}3x3x2−9x
Factor the numerator:
3x(x−3)3x=x−3\frac{3x(x - 3)}{3x} = x - 33x3x(x−3)=x−3
62. Simplify:
4y2−16y4y\frac{4y^2 - 16y}{4y}4y4y2−16y
Factor the numerator:
4y(y−4)4y=y−4\frac{4y(y - 4)}{4y} = y - 44y4y(y−4)=y−4
63. Simplify:
2z2+8z2z\frac{2z^2 + 8z}{2z}2z2z2+8z
Factor the numerator:
2z(z+4)2z=z+4\frac{2z(z + 4)}{2z} = z + 42z2z(z+4)=z+4
64. Expand:
(2x−3)2(2x - 3)^2(2x−3)2
Use the binomial expansion formula (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab +
b^2(a−b)2=a2−2ab+b2:
(2x−3)2=4x2−12x+9(2x - 3)^2 = 4x^2 - 12x + 9(2x−3)2=4x2−12x+9
65. Expand:
(y+4)2(y + 4)^2(y+4)2
Use the binomial expansion formula (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab +
b^2(a+b)2=a2+2ab+b2:
(y+4)2=y2+8y+16(y + 4)^2 = y^2 + 8y + 16(y+4)2=y2+8y+16
66. Expand:
(3a−2b)(3a+2b)(3a - 2b)(3a + 2b)(3a−2b)(3a+2b)
Use the difference of squares formula (a−b)(a+b)=a2−b2(a - b)(a + b) = a^2 - b^2(a−b)
(a+b)=a2−b2:
(3a−2b)(3a+2b)=9a2−4b2(3a - 2b)(3a + 2b) = 9a^2 - 4b^2(3a−2b)(3a+2b)=9a2−4b2
67. Factor:
x2−9x^2 - 9x2−9
This is a difference of squares:
5. Solving the equation:
3z−14+z+23=5\{3z-1}{4} + \{z+2}{3} = 543z−1+3z+2=5
Find a common denominator (12):
3(3z−1)12+4(z+2)12=59z−3+4z+812=513z+512=513z+5=6013z=55z=5513≈4.23\{3(3z-1)}
{12} + \{4(z+2)}{12} = 5 \\ \{9z - 3 + 4z + 8}{12} = 5 \\ \{13z + 5}{12} = 5 \\ 13z + 5 =
60 \\ 13z = 55 \\ z = \{55}{13} \approx 4.23123(3z−1)+124(z+2)=5129z−3+4z+8=51213z+5
=513z+5=6013z=55z=1355≈4.23
6. Solving the equation:
2(a−3)+3(a+4)=52(a-3) + 3(a+4) = 52(a−3)+3(a+4)=5
Expand and combine like terms:
2a−6+3a+12=55a+6=55a=−1a=−152a - 6 + 3a + 12 = 5 \\ 5a + 6 = 5 \\ 5a = -1 \\ a = -\{1}
{5}2a−6+3a+12=55a+6=55a=−1a=−51
7. Solving the equation:
5(b+2)−2(b−3)=105(b+2) - 2(b-3) = 105(b+2)−2(b−3)=10
Expand and combine like terms:
5b+10−2b+6=103b+16=103b=−6b=−25b + 10 - 2b + 6 = 10 \\ 3b + 16 = 10 \\ 3b = -6 \\ b = -
25b+10−2b+6=103b+16=103b=−6b=−2
8. Solving the equation:
(x+1)2−4(x−2)=0(x+1)^2 - 4(x-2) = 0(x+1)2−4(x−2)=0
Expand and simplify:
(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0(x+1)^2 - 4x + 8 = 0 \\ x^2 + 2x + 1 - 4x + 8 =
0 \\ x^2 - 2x + 9 = 0(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0
Solve using the quadratic formula x=−b±b2−4ac2ax = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}x=2a−b±b2−4ac:
x=2±4−362x=2±−322x=1±i8x = \{2 \pm \sqrt{4 - 36}}{2} \\ x = \{2 \pm \sqrt{-32}}{2} \\ x
= 1 \pm i\sqrt{8}x=22±4−36x=22±−32x=1±i8
So, x=1+22ix = 1 + 2\sqrt{2}ix=1+22i and x=1−22ix = 1 - 2\sqrt{2}ix=1−22i.
9. Solving the equation:
3(y−1)2+2(y+3)=03(y-1)^2 + 2(y+3) = 03(y−1)2+2(y+3)=0
Expand and combine like terms:
3(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=03(y^2 - 2y + 1) + 2y + 6 = 0 \\ 3y^2 - 6y
+ 3 + 2y + 6 = 0 \\ 3y^2 - 4y + 9 = 03(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=0
Solve using the quadratic formula y=−b±b2−4ac2ay = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}y=2a−b±b2−4ac:
y=4±16−1086y=4±−926y=4±2i236y=2±i233y = \{4 \pm \sqrt{16 - 108}}{6} \\ y = \{4 \pm \
sqrt{-92}}{6} \\ y = \{4 \pm 2i\sqrt{23}}{6} \\ y = \{2 \pm i\sqrt{23}}{3}y=64±16−108
y=64±−92y=64±2i23y=32±i23
So, y=2+i233y = \{2 + i\sqrt{23}}{3}y=32+i23 and y=2−i233y = \{2 - i\sqrt{23}}
{3}y=32−i23.
10. Solving the equation:
(z−2)2−5(z+1)=0(z-2)^2 - 5(z+1) = 0(z−2)2−5(z+1)=0
Expand and simplify:
z2−4z+4−5z−5=0z2−9z−1=0z^2 - 4z + 4 - 5z - 5 = 0 \\ z^2 - 9z - 1 =
0z2−4z+4−5z−5=0z2−9z−1=0
Solve using the quadratic formula z=−b±b2−4ac2az = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}z=2a−b±b2−4ac:
z=9±81+42z=9±852z = \{9 \pm \sqrt{81 + 4}}{2} \\ z = \{9 \pm \sqrt{85}}{2}z=29±81+4
z=29±85
So, z=9+852z = \{9 + \sqrt{85}}{2}z=29+85 and z=9−852z = \{9 - \sqrt{85}}{2}z=29−85.
Advanced Algebraic Expressions
61. Simplify:
3x2−9x3x\frac{3x^2 - 9x}{3x}3x3x2−9x
Factor the numerator:
3x(x−3)3x=x−3\frac{3x(x - 3)}{3x} = x - 33x3x(x−3)=x−3
62. Simplify:
4y2−16y4y\frac{4y^2 - 16y}{4y}4y4y2−16y
Factor the numerator:
4y(y−4)4y=y−4\frac{4y(y - 4)}{4y} = y - 44y4y(y−4)=y−4
63. Simplify:
2z2+8z2z\frac{2z^2 + 8z}{2z}2z2z2+8z
Factor the numerator:
2z(z+4)2z=z+4\frac{2z(z + 4)}{2z} = z + 42z2z(z+4)=z+4
64. Expand:
(2x−3)2(2x - 3)^2(2x−3)2
Use the binomial expansion formula (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab +
b^2(a−b)2=a2−2ab+b2:
(2x−3)2=4x2−12x+9(2x - 3)^2 = 4x^2 - 12x + 9(2x−3)2=4x2−12x+9
65. Expand:
(y+4)2(y + 4)^2(y+4)2
Use the binomial expansion formula (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab +
b^2(a+b)2=a2+2ab+b2:
(y+4)2=y2+8y+16(y + 4)^2 = y^2 + 8y + 16(y+4)2=y2+8y+16
66. Expand:
(3a−2b)(3a+2b)(3a - 2b)(3a + 2b)(3a−2b)(3a+2b)
Use the difference of squares formula (a−b)(a+b)=a2−b2(a - b)(a + b) = a^2 - b^2(a−b)
(a+b)=a2−b2:
(3a−2b)(3a+2b)=9a2−4b2(3a - 2b)(3a + 2b) = 9a^2 - 4b^2(3a−2b)(3a+2b)=9a2−4b2
67. Factor:
x2−9x^2 - 9x2−9
This is a difference of squares:
5. Solving the equation:
3z−14+z+23=5\{3z-1}{4} + \{z+2}{3} = 543z−1+3z+2=5
Find a common denominator (12):
3(3z−1)12+4(z+2)12=59z−3+4z+812=513z+512=513z+5=6013z=55z=5513≈4.23\{3(3z-1)}
{12} + \{4(z+2)}{12} = 5 \\ \{9z - 3 + 4z + 8}{12} = 5 \\ \{13z + 5}{12} = 5 \\ 13z + 5 =
60 \\ 13z = 55 \\ z = \{55}{13} \approx 4.23123(3z−1)+124(z+2)=5129z−3+4z+8=51213z+5
=513z+5=6013z=55z=1355≈4.23
6. Solving the equation:
2(a−3)+3(a+4)=52(a-3) + 3(a+4) = 52(a−3)+3(a+4)=5
Expand and combine like terms:
2a−6+3a+12=55a+6=55a=−1a=−152a - 6 + 3a + 12 = 5 \\ 5a + 6 = 5 \\ 5a = -1 \\ a = -\{1}
{5}2a−6+3a+12=55a+6=55a=−1a=−51
7. Solving the equation:
5(b+2)−2(b−3)=105(b+2) - 2(b-3) = 105(b+2)−2(b−3)=10
Expand and combine like terms:
5b+10−2b+6=103b+16=103b=−6b=−25b + 10 - 2b + 6 = 10 \\ 3b + 16 = 10 \\ 3b = -6 \\ b = -
25b+10−2b+6=103b+16=103b=−6b=−2
8. Solving the equation:
(x+1)2−4(x−2)=0(x+1)^2 - 4(x-2) = 0(x+1)2−4(x−2)=0
Expand and simplify:
(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0(x+1)^2 - 4x + 8 = 0 \\ x^2 + 2x + 1 - 4x + 8 =
0 \\ x^2 - 2x + 9 = 0(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0
Solve using the quadratic formula x=−b±b2−4ac2ax = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}x=2a−b±b2−4ac:
x=2±4−362x=2±−322x=1±i8x = \{2 \pm \sqrt{4 - 36}}{2} \\ x = \{2 \pm \sqrt{-32}}{2} \\ x
= 1 \pm i\sqrt{8}x=22±4−36x=22±−32x=1±i8
So, x=1+22ix = 1 + 2\sqrt{2}ix=1+22i and x=1−22ix = 1 - 2\sqrt{2}ix=1−22i.
9. Solving the equation:
3(y−1)2+2(y+3)=03(y-1)^2 + 2(y+3) = 03(y−1)2+2(y+3)=0
Expand and combine like terms:
3(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=03(y^2 - 2y + 1) + 2y + 6 = 0 \\ 3y^2 - 6y
+ 3 + 2y + 6 = 0 \\ 3y^2 - 4y + 9 = 03(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=0
Solve using the quadratic formula y=−b±b2−4ac2ay = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}y=2a−b±b2−4ac:
y=4±16−1086y=4±−926y=4±2i236y=2±i233y = \{4 \pm \sqrt{16 - 108}}{6} \\ y = \{4 \pm \
sqrt{-92}}{6} \\ y = \{4 \pm 2i\sqrt{23}}{6} \\ y = \{2 \pm i\sqrt{23}}{3}y=64±16−108
y=64±−92y=64±2i23y=32±i23
So, y=2+i233y = \{2 + i\sqrt{23}}{3}y=32+i23 and y=2−i233y = \{2 - i\sqrt{23}}
{3}y=32−i23.
10. Solving the equation:
(z−2)2−5(z+1)=0(z-2)^2 - 5(z+1) = 0(z−2)2−5(z+1)=0
Expand and simplify:
z2−4z+4−5z−5=0z2−9z−1=0z^2 - 4z + 4 - 5z - 5 = 0 \\ z^2 - 9z - 1 =
0z2−4z+4−5z−5=0z2−9z−1=0
Solve using the quadratic formula z=−b±b2−4ac2az = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}z=2a−b±b2−4ac:
z=9±81+42z=9±852z = \{9 \pm \sqrt{81 + 4}}{2} \\ z = \{9 \pm \sqrt{85}}{2}z=29±81+4
z=29±85
So, z=9+852z = \{9 + \sqrt{85}}{2}z=29+85 and z=9−852z = \{9 - \sqrt{85}}{2}z=29−85.
Advanced Algebraic Expressions
61. Simplify:
3x2−9x3x\frac{3x^2 - 9x}{3x}3x3x2−9x
Factor the numerator:
3x(x−3)3x=x−3\frac{3x(x - 3)}{3x} = x - 33x3x(x−3)=x−3
62. Simplify:
4y2−16y4y\frac{4y^2 - 16y}{4y}4y4y2−16y
Factor the numerator:
4y(y−4)4y=y−4\frac{4y(y - 4)}{4y} = y - 44y4y(y−4)=y−4
63. Simplify:
2z2+8z2z\frac{2z^2 + 8z}{2z}2z2z2+8z
Factor the numerator:
2z(z+4)2z=z+4\frac{2z(z + 4)}{2z} = z + 42z2z(z+4)=z+4
64. Expand:
(2x−3)2(2x - 3)^2(2x−3)2
Use the binomial expansion formula (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab +
b^2(a−b)2=a2−2ab+b2:
(2x−3)2=4x2−12x+9(2x - 3)^2 = 4x^2 - 12x + 9(2x−3)2=4x2−12x+9
65. Expand:
(y+4)2(y + 4)^2(y+4)2
Use the binomial expansion formula (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab +
b^2(a+b)2=a2+2ab+b2:
(y+4)2=y2+8y+16(y + 4)^2 = y^2 + 8y + 16(y+4)2=y2+8y+16
66. Expand:
(3a−2b)(3a+2b)(3a - 2b)(3a + 2b)(3a−2b)(3a+2b)
Use the difference of squares formula (a−b)(a+b)=a2−b2(a - b)(a + b) = a^2 - b^2(a−b)
(a+b)=a2−b2:
(3a−2b)(3a+2b)=9a2−4b2(3a - 2b)(3a + 2b) = 9a^2 - 4b^2(3a−2b)(3a+2b)=9a2−4b2
67. Factor:
x2−9x^2 - 9x2−9
This is a difference of squares:
5. Solving the equation:
3z−14+z+23=5\{3z-1}{4} + \{z+2}{3} = 543z−1+3z+2=5
Find a common denominator (12):
3(3z−1)12+4(z+2)12=59z−3+4z+812=513z+512=513z+5=6013z=55z=5513≈4.23\{3(3z-1)}
{12} + \{4(z+2)}{12} = 5 \\ \{9z - 3 + 4z + 8}{12} = 5 \\ \{13z + 5}{12} = 5 \\ 13z + 5 =
60 \\ 13z = 55 \\ z = \{55}{13} \approx 4.23123(3z−1)+124(z+2)=5129z−3+4z+8=51213z+5
=513z+5=6013z=55z=1355≈4.23
6. Solving the equation:
2(a−3)+3(a+4)=52(a-3) + 3(a+4) = 52(a−3)+3(a+4)=5
Expand and combine like terms:
2a−6+3a+12=55a+6=55a=−1a=−152a - 6 + 3a + 12 = 5 \\ 5a + 6 = 5 \\ 5a = -1 \\ a = -\{1}
{5}2a−6+3a+12=55a+6=55a=−1a=−51
7. Solving the equation:
5(b+2)−2(b−3)=105(b+2) - 2(b-3) = 105(b+2)−2(b−3)=10
Expand and combine like terms:
5b+10−2b+6=103b+16=103b=−6b=−25b + 10 - 2b + 6 = 10 \\ 3b + 16 = 10 \\ 3b = -6 \\ b = -
25b+10−2b+6=103b+16=103b=−6b=−2
8. Solving the equation:
(x+1)2−4(x−2)=0(x+1)^2 - 4(x-2) = 0(x+1)2−4(x−2)=0
Expand and simplify:
(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0(x+1)^2 - 4x + 8 = 0 \\ x^2 + 2x + 1 - 4x + 8 =
0 \\ x^2 - 2x + 9 = 0(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0
Solve using the quadratic formula x=−b±b2−4ac2ax = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}x=2a−b±b2−4ac:
x=2±4−362x=2±−322x=1±i8x = \{2 \pm \sqrt{4 - 36}}{2} \\ x = \{2 \pm \sqrt{-32}}{2} \\ x
= 1 \pm i\sqrt{8}x=22±4−36x=22±−32x=1±i8
So, x=1+22ix = 1 + 2\sqrt{2}ix=1+22i and x=1−22ix = 1 - 2\sqrt{2}ix=1−22i.
9. Solving the equation:
3(y−1)2+2(y+3)=03(y-1)^2 + 2(y+3) = 03(y−1)2+2(y+3)=0
Expand and combine like terms:
3(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=03(y^2 - 2y + 1) + 2y + 6 = 0 \\ 3y^2 - 6y
+ 3 + 2y + 6 = 0 \\ 3y^2 - 4y + 9 = 03(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=0
Solve using the quadratic formula y=−b±b2−4ac2ay = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}y=2a−b±b2−4ac:
y=4±16−1086y=4±−926y=4±2i236y=2±i233y = \{4 \pm \sqrt{16 - 108}}{6} \\ y = \{4 \pm \
sqrt{-92}}{6} \\ y = \{4 \pm 2i\sqrt{23}}{6} \\ y = \{2 \pm i\sqrt{23}}{3}y=64±16−108
y=64±−92y=64±2i23y=32±i23
So, y=2+i233y = \{2 + i\sqrt{23}}{3}y=32+i23 and y=2−i233y = \{2 - i\sqrt{23}}
{3}y=32−i23.
10. Solving the equation:
(z−2)2−5(z+1)=0(z-2)^2 - 5(z+1) = 0(z−2)2−5(z+1)=0
Expand and simplify:
z2−4z+4−5z−5=0z2−9z−1=0z^2 - 4z + 4 - 5z - 5 = 0 \\ z^2 - 9z - 1 =
0z2−4z+4−5z−5=0z2−9z−1=0
Solve using the quadratic formula z=−b±b2−4ac2az = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}z=2a−b±b2−4ac:
z=9±81+42z=9±852z = \{9 \pm \sqrt{81 + 4}}{2} \\ z = \{9 \pm \sqrt{85}}{2}z=29±81+4
z=29±85
So, z=9+852z = \{9 + \sqrt{85}}{2}z=29+85 and z=9−852z = \{9 - \sqrt{85}}{2}z=29−85.
Advanced Algebraic Expressions
61. Simplify:
3x2−9x3x\frac{3x^2 - 9x}{3x}3x3x2−9x
Factor the numerator:
3x(x−3)3x=x−3\frac{3x(x - 3)}{3x} = x - 33x3x(x−3)=x−3
62. Simplify:
4y2−16y4y\frac{4y^2 - 16y}{4y}4y4y2−16y
Factor the numerator:
4y(y−4)4y=y−4\frac{4y(y - 4)}{4y} = y - 44y4y(y−4)=y−4
63. Simplify:
2z2+8z2z\frac{2z^2 + 8z}{2z}2z2z2+8z
Factor the numerator:
2z(z+4)2z=z+4\frac{2z(z + 4)}{2z} = z + 42z2z(z+4)=z+4
64. Expand:
(2x−3)2(2x - 3)^2(2x−3)2
Use the binomial expansion formula (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab +
b^2(a−b)2=a2−2ab+b2:
(2x−3)2=4x2−12x+9(2x - 3)^2 = 4x^2 - 12x + 9(2x−3)2=4x2−12x+9
65. Expand:
(y+4)2(y + 4)^2(y+4)2
Use the binomial expansion formula (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab +
b^2(a+b)2=a2+2ab+b2:
(y+4)2=y2+8y+16(y + 4)^2 = y^2 + 8y + 16(y+4)2=y2+8y+16
66. Expand:
(3a−2b)(3a+2b)(3a - 2b)(3a + 2b)(3a−2b)(3a+2b)
Use the difference of squares formula (a−b)(a+b)=a2−b2(a - b)(a + b) = a^2 - b^2(a−b)
(a+b)=a2−b2:
(3a−2b)(3a+2b)=9a2−4b2(3a - 2b)(3a + 2b) = 9a^2 - 4b^2(3a−2b)(3a+2b)=9a2−4b2
67. Factor:
x2−9x^2 - 9x2−9
This is a difference of squares:
5. Solving the equation:
3z−14+z+23=5\{3z-1}{4} + \{z+2}{3} = 543z−1+3z+2=5
Find a common denominator (12):
3(3z−1)12+4(z+2)12=59z−3+4z+812=513z+512=513z+5=6013z=55z=5513≈4.23\{3(3z-1)}
{12} + \{4(z+2)}{12} = 5 \\ \{9z - 3 + 4z + 8}{12} = 5 \\ \{13z + 5}{12} = 5 \\ 13z + 5 =
60 \\ 13z = 55 \\ z = \{55}{13} \approx 4.23123(3z−1)+124(z+2)=5129z−3+4z+8=51213z+5
=513z+5=6013z=55z=1355≈4.23
6. Solving the equation:
2(a−3)+3(a+4)=52(a-3) + 3(a+4) = 52(a−3)+3(a+4)=5
Expand and combine like terms:
2a−6+3a+12=55a+6=55a=−1a=−152a - 6 + 3a + 12 = 5 \\ 5a + 6 = 5 \\ 5a = -1 \\ a = -\{1}
{5}2a−6+3a+12=55a+6=55a=−1a=−51
7. Solving the equation:
5(b+2)−2(b−3)=105(b+2) - 2(b-3) = 105(b+2)−2(b−3)=10
Expand and combine like terms:
5b+10−2b+6=103b+16=103b=−6b=−25b + 10 - 2b + 6 = 10 \\ 3b + 16 = 10 \\ 3b = -6 \\ b = -
25b+10−2b+6=103b+16=103b=−6b=−2
8. Solving the equation:
(x+1)2−4(x−2)=0(x+1)^2 - 4(x-2) = 0(x+1)2−4(x−2)=0
Expand and simplify:
(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0(x+1)^2 - 4x + 8 = 0 \\ x^2 + 2x + 1 - 4x + 8 =
0 \\ x^2 - 2x + 9 = 0(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0
Solve using the quadratic formula x=−b±b2−4ac2ax = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}x=2a−b±b2−4ac:
x=2±4−362x=2±−322x=1±i8x = \{2 \pm \sqrt{4 - 36}}{2} \\ x = \{2 \pm \sqrt{-32}}{2} \\ x
= 1 \pm i\sqrt{8}x=22±4−36x=22±−32x=1±i8
So, x=1+22ix = 1 + 2\sqrt{2}ix=1+22i and x=1−22ix = 1 - 2\sqrt{2}ix=1−22i.
9. Solving the equation:
3(y−1)2+2(y+3)=03(y-1)^2 + 2(y+3) = 03(y−1)2+2(y+3)=0
Expand and combine like terms:
3(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=03(y^2 - 2y + 1) + 2y + 6 = 0 \\ 3y^2 - 6y
+ 3 + 2y + 6 = 0 \\ 3y^2 - 4y + 9 = 03(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=0
Solve using the quadratic formula y=−b±b2−4ac2ay = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}y=2a−b±b2−4ac:
y=4±16−1086y=4±−926y=4±2i236y=2±i233y = \{4 \pm \sqrt{16 - 108}}{6} \\ y = \{4 \pm \
sqrt{-92}}{6} \\ y = \{4 \pm 2i\sqrt{23}}{6} \\ y = \{2 \pm i\sqrt{23}}{3}y=64±16−108
y=64±−92y=64±2i23y=32±i23
So, y=2+i233y = \{2 + i\sqrt{23}}{3}y=32+i23 and y=2−i233y = \{2 - i\sqrt{23}}
{3}y=32−i23.
10. Solving the equation:
(z−2)2−5(z+1)=0(z-2)^2 - 5(z+1) = 0(z−2)2−5(z+1)=0
Expand and simplify:
z2−4z+4−5z−5=0z2−9z−1=0z^2 - 4z + 4 - 5z - 5 = 0 \\ z^2 - 9z - 1 =
0z2−4z+4−5z−5=0z2−9z−1=0
Solve using the quadratic formula z=−b±b2−4ac2az = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}z=2a−b±b2−4ac:
z=9±81+42z=9±852z = \{9 \pm \sqrt{81 + 4}}{2} \\ z = \{9 \pm \sqrt{85}}{2}z=29±81+4
z=29±85
So, z=9+852z = \{9 + \sqrt{85}}{2}z=29+85 and z=9−852z = \{9 - \sqrt{85}}{2}z=29−85.
Advanced Algebraic Expressions
61. Simplify:
3x2−9x3x\frac{3x^2 - 9x}{3x}3x3x2−9x
Factor the numerator:
3x(x−3)3x=x−3\frac{3x(x - 3)}{3x} = x - 33x3x(x−3)=x−3
62. Simplify:
4y2−16y4y\frac{4y^2 - 16y}{4y}4y4y2−16y
Factor the numerator:
4y(y−4)4y=y−4\frac{4y(y - 4)}{4y} = y - 44y4y(y−4)=y−4
63. Simplify:
2z2+8z2z\frac{2z^2 + 8z}{2z}2z2z2+8z
Factor the numerator:
2z(z+4)2z=z+4\frac{2z(z + 4)}{2z} = z + 42z2z(z+4)=z+4
64. Expand:
(2x−3)2(2x - 3)^2(2x−3)2
Use the binomial expansion formula (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab +
b^2(a−b)2=a2−2ab+b2:
(2x−3)2=4x2−12x+9(2x - 3)^2 = 4x^2 - 12x + 9(2x−3)2=4x2−12x+9
65. Expand:
(y+4)2(y + 4)^2(y+4)2
Use the binomial expansion formula (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab +
b^2(a+b)2=a2+2ab+b2:
(y+4)2=y2+8y+16(y + 4)^2 = y^2 + 8y + 16(y+4)2=y2+8y+16
66. Expand:
(3a−2b)(3a+2b)(3a - 2b)(3a + 2b)(3a−2b)(3a+2b)
Use the difference of squares formula (a−b)(a+b)=a2−b2(a - b)(a + b) = a^2 - b^2(a−b)
(a+b)=a2−b2:
(3a−2b)(3a+2b)=9a2−4b2(3a - 2b)(3a + 2b) = 9a^2 - 4b^2(3a−2b)(3a+2b)=9a2−4b2
67. Factor:
x2−9x^2 - 9x2−9
This is a difference of squares:
5. Solving the equation:
3z−14+z+23=5\{3z-1}{4} + \{z+2}{3} = 543z−1+3z+2=5
Find a common denominator (12):
3(3z−1)12+4(z+2)12=59z−3+4z+812=513z+512=513z+5=6013z=55z=5513≈4.23\{3(3z-1)}
{12} + \{4(z+2)}{12} = 5 \\ \{9z - 3 + 4z + 8}{12} = 5 \\ \{13z + 5}{12} = 5 \\ 13z + 5 =
60 \\ 13z = 55 \\ z = \{55}{13} \approx 4.23123(3z−1)+124(z+2)=5129z−3+4z+8=51213z+5
=513z+5=6013z=55z=1355≈4.23
6. Solving the equation:
2(a−3)+3(a+4)=52(a-3) + 3(a+4) = 52(a−3)+3(a+4)=5
Expand and combine like terms:
2a−6+3a+12=55a+6=55a=−1a=−152a - 6 + 3a + 12 = 5 \\ 5a + 6 = 5 \\ 5a = -1 \\ a = -\{1}
{5}2a−6+3a+12=55a+6=55a=−1a=−51
7. Solving the equation:
5(b+2)−2(b−3)=105(b+2) - 2(b-3) = 105(b+2)−2(b−3)=10
Expand and combine like terms:
5b+10−2b+6=103b+16=103b=−6b=−25b + 10 - 2b + 6 = 10 \\ 3b + 16 = 10 \\ 3b = -6 \\ b = -
25b+10−2b+6=103b+16=103b=−6b=−2
8. Solving the equation:
(x+1)2−4(x−2)=0(x+1)^2 - 4(x-2) = 0(x+1)2−4(x−2)=0
Expand and simplify:
(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0(x+1)^2 - 4x + 8 = 0 \\ x^2 + 2x + 1 - 4x + 8 =
0 \\ x^2 - 2x + 9 = 0(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0
Solve using the quadratic formula x=−b±b2−4ac2ax = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}x=2a−b±b2−4ac:
x=2±4−362x=2±−322x=1±i8x = \{2 \pm \sqrt{4 - 36}}{2} \\ x = \{2 \pm \sqrt{-32}}{2} \\ x
= 1 \pm i\sqrt{8}x=22±4−36x=22±−32x=1±i8
So, x=1+22ix = 1 + 2\sqrt{2}ix=1+22i and x=1−22ix = 1 - 2\sqrt{2}ix=1−22i.
9. Solving the equation:
3(y−1)2+2(y+3)=03(y-1)^2 + 2(y+3) = 03(y−1)2+2(y+3)=0
Expand and combine like terms:
3(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=03(y^2 - 2y + 1) + 2y + 6 = 0 \\ 3y^2 - 6y
+ 3 + 2y + 6 = 0 \\ 3y^2 - 4y + 9 = 03(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=0
Solve using the quadratic formula y=−b±b2−4ac2ay = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}y=2a−b±b2−4ac:
y=4±16−1086y=4±−926y=4±2i236y=2±i233y = \{4 \pm \sqrt{16 - 108}}{6} \\ y = \{4 \pm \
sqrt{-92}}{6} \\ y = \{4 \pm 2i\sqrt{23}}{6} \\ y = \{2 \pm i\sqrt{23}}{3}y=64±16−108
y=64±−92y=64±2i23y=32±i23
So, y=2+i233y = \{2 + i\sqrt{23}}{3}y=32+i23 and y=2−i233y = \{2 - i\sqrt{23}}
{3}y=32−i23.
10. Solving the equation:
(z−2)2−5(z+1)=0(z-2)^2 - 5(z+1) = 0(z−2)2−5(z+1)=0
Expand and simplify:
z2−4z+4−5z−5=0z2−9z−1=0z^2 - 4z + 4 - 5z - 5 = 0 \\ z^2 - 9z - 1 =
0z2−4z+4−5z−5=0z2−9z−1=0
Solve using the quadratic formula z=−b±b2−4ac2az = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}z=2a−b±b2−4ac:
z=9±81+42z=9±852z = \{9 \pm \sqrt{81 + 4}}{2} \\ z = \{9 \pm \sqrt{85}}{2}z=29±81+4
z=29±85
So, z=9+852z = \{9 + \sqrt{85}}{2}z=29+85 and z=9−852z = \{9 - \sqrt{85}}{2}z=29−85.
Advanced Algebraic Expressions
61. Simplify:
3x2−9x3x\frac{3x^2 - 9x}{3x}3x3x2−9x
Factor the numerator:
3x(x−3)3x=x−3\frac{3x(x - 3)}{3x} = x - 33x3x(x−3)=x−3
62. Simplify:
4y2−16y4y\frac{4y^2 - 16y}{4y}4y4y2−16y
Factor the numerator:
4y(y−4)4y=y−4\frac{4y(y - 4)}{4y} = y - 44y4y(y−4)=y−4
63. Simplify:
2z2+8z2z\frac{2z^2 + 8z}{2z}2z2z2+8z
Factor the numerator:
2z(z+4)2z=z+4\frac{2z(z + 4)}{2z} = z + 42z2z(z+4)=z+4
64. Expand:
(2x−3)2(2x - 3)^2(2x−3)2
Use the binomial expansion formula (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab +
b^2(a−b)2=a2−2ab+b2:
(2x−3)2=4x2−12x+9(2x - 3)^2 = 4x^2 - 12x + 9(2x−3)2=4x2−12x+9
65. Expand:
(y+4)2(y + 4)^2(y+4)2
Use the binomial expansion formula (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab +
b^2(a+b)2=a2+2ab+b2:
(y+4)2=y2+8y+16(y + 4)^2 = y^2 + 8y + 16(y+4)2=y2+8y+16
66. Expand:
(3a−2b)(3a+2b)(3a - 2b)(3a + 2b)(3a−2b)(3a+2b)
Use the difference of squares formula (a−b)(a+b)=a2−b2(a - b)(a + b) = a^2 - b^2(a−b)
(a+b)=a2−b2:
(3a−2b)(3a+2b)=9a2−4b2(3a - 2b)(3a + 2b) = 9a^2 - 4b^2(3a−2b)(3a+2b)=9a2−4b2
67. Factor:
x2−9x^2 - 9x2−9
This is a difference of squares:
5. Solving the equation:
3z−14+z+23=5\{3z-1}{4} + \{z+2}{3} = 543z−1+3z+2=5
Find a common denominator (12):
3(3z−1)12+4(z+2)12=59z−3+4z+812=513z+512=513z+5=6013z=55z=5513≈4.23\{3(3z-1)}
{12} + \{4(z+2)}{12} = 5 \\ \{9z - 3 + 4z + 8}{12} = 5 \\ \{13z + 5}{12} = 5 \\ 13z + 5 =
60 \\ 13z = 55 \\ z = \{55}{13} \approx 4.23123(3z−1)+124(z+2)=5129z−3+4z+8=51213z+5
=513z+5=6013z=55z=1355≈4.23
6. Solving the equation:
2(a−3)+3(a+4)=52(a-3) + 3(a+4) = 52(a−3)+3(a+4)=5
Expand and combine like terms:
2a−6+3a+12=55a+6=55a=−1a=−152a - 6 + 3a + 12 = 5 \\ 5a + 6 = 5 \\ 5a = -1 \\ a = -\{1}
{5}2a−6+3a+12=55a+6=55a=−1a=−51
7. Solving the equation:
5(b+2)−2(b−3)=105(b+2) - 2(b-3) = 105(b+2)−2(b−3)=10
Expand and combine like terms:
5b+10−2b+6=103b+16=103b=−6b=−25b + 10 - 2b + 6 = 10 \\ 3b + 16 = 10 \\ 3b = -6 \\ b = -
25b+10−2b+6=103b+16=103b=−6b=−2
8. Solving the equation:
(x+1)2−4(x−2)=0(x+1)^2 - 4(x-2) = 0(x+1)2−4(x−2)=0
Expand and simplify:
(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0(x+1)^2 - 4x + 8 = 0 \\ x^2 + 2x + 1 - 4x + 8 =
0 \\ x^2 - 2x + 9 = 0(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0
Solve using the quadratic formula x=−b±b2−4ac2ax = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}x=2a−b±b2−4ac:
x=2±4−362x=2±−322x=1±i8x = \{2 \pm \sqrt{4 - 36}}{2} \\ x = \{2 \pm \sqrt{-32}}{2} \\ x
= 1 \pm i\sqrt{8}x=22±4−36x=22±−32x=1±i8
So, x=1+22ix = 1 + 2\sqrt{2}ix=1+22i and x=1−22ix = 1 - 2\sqrt{2}ix=1−22i.
9. Solving the equation:
3(y−1)2+2(y+3)=03(y-1)^2 + 2(y+3) = 03(y−1)2+2(y+3)=0
Expand and combine like terms:
3(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=03(y^2 - 2y + 1) + 2y + 6 = 0 \\ 3y^2 - 6y
+ 3 + 2y + 6 = 0 \\ 3y^2 - 4y + 9 = 03(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=0
Solve using the quadratic formula y=−b±b2−4ac2ay = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}y=2a−b±b2−4ac:
y=4±16−1086y=4±−926y=4±2i236y=2±i233y = \{4 \pm \sqrt{16 - 108}}{6} \\ y = \{4 \pm \
sqrt{-92}}{6} \\ y = \{4 \pm 2i\sqrt{23}}{6} \\ y = \{2 \pm i\sqrt{23}}{3}y=64±16−108
y=64±−92y=64±2i23y=32±i23
So, y=2+i233y = \{2 + i\sqrt{23}}{3}y=32+i23 and y=2−i233y = \{2 - i\sqrt{23}}
{3}y=32−i23.
10. Solving the equation:
(z−2)2−5(z+1)=0(z-2)^2 - 5(z+1) = 0(z−2)2−5(z+1)=0
Expand and simplify:
z2−4z+4−5z−5=0z2−9z−1=0z^2 - 4z + 4 - 5z - 5 = 0 \\ z^2 - 9z - 1 =
0z2−4z+4−5z−5=0z2−9z−1=0
Solve using the quadratic formula z=−b±b2−4ac2az = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}z=2a−b±b2−4ac:
z=9±81+42z=9±852z = \{9 \pm \sqrt{81 + 4}}{2} \\ z = \{9 \pm \sqrt{85}}{2}z=29±81+4
z=29±85
So, z=9+852z = \{9 + \sqrt{85}}{2}z=29+85 and z=9−852z = \{9 - \sqrt{85}}{2}z=29−85.
Advanced Algebraic Expressions
61. Simplify:
3x2−9x3x\frac{3x^2 - 9x}{3x}3x3x2−9x
Factor the numerator:
3x(x−3)3x=x−3\frac{3x(x - 3)}{3x} = x - 33x3x(x−3)=x−3
62. Simplify:
4y2−16y4y\frac{4y^2 - 16y}{4y}4y4y2−16y
Factor the numerator:
4y(y−4)4y=y−4\frac{4y(y - 4)}{4y} = y - 44y4y(y−4)=y−4
63. Simplify:
2z2+8z2z\frac{2z^2 + 8z}{2z}2z2z2+8z
Factor the numerator:
2z(z+4)2z=z+4\frac{2z(z + 4)}{2z} = z + 42z2z(z+4)=z+4
64. Expand:
(2x−3)2(2x - 3)^2(2x−3)2
Use the binomial expansion formula (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab +
b^2(a−b)2=a2−2ab+b2:
(2x−3)2=4x2−12x+9(2x - 3)^2 = 4x^2 - 12x + 9(2x−3)2=4x2−12x+9
65. Expand:
(y+4)2(y + 4)^2(y+4)2
Use the binomial expansion formula (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab +
b^2(a+b)2=a2+2ab+b2:
(y+4)2=y2+8y+16(y + 4)^2 = y^2 + 8y + 16(y+4)2=y2+8y+16
66. Expand:
(3a−2b)(3a+2b)(3a - 2b)(3a + 2b)(3a−2b)(3a+2b)
Use the difference of squares formula (a−b)(a+b)=a2−b2(a - b)(a + b) = a^2 - b^2(a−b)
(a+b)=a2−b2:
(3a−2b)(3a+2b)=9a2−4b2(3a - 2b)(3a + 2b) = 9a^2 - 4b^2(3a−2b)(3a+2b)=9a2−4b2
67. Factor:
x2−9x^2 - 9x2−9
This is a difference of squares:
5. Solving the equation:
3z−14+z+23=5\{3z-1}{4} + \{z+2}{3} = 543z−1+3z+2=5
Find a common denominator (12):
3(3z−1)12+4(z+2)12=59z−3+4z+812=513z+512=513z+5=6013z=55z=5513≈4.23\{3(3z-1)}
{12} + \{4(z+2)}{12} = 5 \\ \{9z - 3 + 4z + 8}{12} = 5 \\ \{13z + 5}{12} = 5 \\ 13z + 5 =
60 \\ 13z = 55 \\ z = \{55}{13} \approx 4.23123(3z−1)+124(z+2)=5129z−3+4z+8=51213z+5
=513z+5=6013z=55z=1355≈4.23
6. Solving the equation:
2(a−3)+3(a+4)=52(a-3) + 3(a+4) = 52(a−3)+3(a+4)=5
Expand and combine like terms:
2a−6+3a+12=55a+6=55a=−1a=−152a - 6 + 3a + 12 = 5 \\ 5a + 6 = 5 \\ 5a = -1 \\ a = -\{1}
{5}2a−6+3a+12=55a+6=55a=−1a=−51
7. Solving the equation:
5(b+2)−2(b−3)=105(b+2) - 2(b-3) = 105(b+2)−2(b−3)=10
Expand and combine like terms:
5b+10−2b+6=103b+16=103b=−6b=−25b + 10 - 2b + 6 = 10 \\ 3b + 16 = 10 \\ 3b = -6 \\ b = -
25b+10−2b+6=103b+16=103b=−6b=−2
8. Solving the equation:
(x+1)2−4(x−2)=0(x+1)^2 - 4(x-2) = 0(x+1)2−4(x−2)=0
Expand and simplify:
(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0(x+1)^2 - 4x + 8 = 0 \\ x^2 + 2x + 1 - 4x + 8 =
0 \\ x^2 - 2x + 9 = 0(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0
Solve using the quadratic formula x=−b±b2−4ac2ax = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}x=2a−b±b2−4ac:
x=2±4−362x=2±−322x=1±i8x = \{2 \pm \sqrt{4 - 36}}{2} \\ x = \{2 \pm \sqrt{-32}}{2} \\ x
= 1 \pm i\sqrt{8}x=22±4−36x=22±−32x=1±i8
So, x=1+22ix = 1 + 2\sqrt{2}ix=1+22i and x=1−22ix = 1 - 2\sqrt{2}ix=1−22i.
9. Solving the equation:
3(y−1)2+2(y+3)=03(y-1)^2 + 2(y+3) = 03(y−1)2+2(y+3)=0
Expand and combine like terms:
3(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=03(y^2 - 2y + 1) + 2y + 6 = 0 \\ 3y^2 - 6y
+ 3 + 2y + 6 = 0 \\ 3y^2 - 4y + 9 = 03(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=0
Solve using the quadratic formula y=−b±b2−4ac2ay = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}y=2a−b±b2−4ac:
y=4±16−1086y=4±−926y=4±2i236y=2±i233y = \{4 \pm \sqrt{16 - 108}}{6} \\ y = \{4 \pm \
sqrt{-92}}{6} \\ y = \{4 \pm 2i\sqrt{23}}{6} \\ y = \{2 \pm i\sqrt{23}}{3}y=64±16−108
y=64±−92y=64±2i23y=32±i23
So, y=2+i233y = \{2 + i\sqrt{23}}{3}y=32+i23 and y=2−i233y = \{2 - i\sqrt{23}}
{3}y=32−i23.
10. Solving the equation:
(z−2)2−5(z+1)=0(z-2)^2 - 5(z+1) = 0(z−2)2−5(z+1)=0
Expand and simplify:
z2−4z+4−5z−5=0z2−9z−1=0z^2 - 4z + 4 - 5z - 5 = 0 \\ z^2 - 9z - 1 =
0z2−4z+4−5z−5=0z2−9z−1=0
Solve using the quadratic formula z=−b±b2−4ac2az = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}z=2a−b±b2−4ac:
z=9±81+42z=9±852z = \{9 \pm \sqrt{81 + 4}}{2} \\ z = \{9 \pm \sqrt{85}}{2}z=29±81+4
z=29±85
So, z=9+852z = \{9 + \sqrt{85}}{2}z=29+85 and z=9−852z = \{9 - \sqrt{85}}{2}z=29−85.
Advanced Algebraic Expressions
61. Simplify:
3x2−9x3x\frac{3x^2 - 9x}{3x}3x3x2−9x
Factor the numerator:
3x(x−3)3x=x−3\frac{3x(x - 3)}{3x} = x - 33x3x(x−3)=x−3
62. Simplify:
4y2−16y4y\frac{4y^2 - 16y}{4y}4y4y2−16y
Factor the numerator:
4y(y−4)4y=y−4\frac{4y(y - 4)}{4y} = y - 44y4y(y−4)=y−4
63. Simplify:
2z2+8z2z\frac{2z^2 + 8z}{2z}2z2z2+8z
Factor the numerator:
2z(z+4)2z=z+4\frac{2z(z + 4)}{2z} = z + 42z2z(z+4)=z+4
64. Expand:
(2x−3)2(2x - 3)^2(2x−3)2
Use the binomial expansion formula (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab +
b^2(a−b)2=a2−2ab+b2:
(2x−3)2=4x2−12x+9(2x - 3)^2 = 4x^2 - 12x + 9(2x−3)2=4x2−12x+9
65. Expand:
(y+4)2(y + 4)^2(y+4)2
Use the binomial expansion formula (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab +
b^2(a+b)2=a2+2ab+b2:
(y+4)2=y2+8y+16(y + 4)^2 = y^2 + 8y + 16(y+4)2=y2+8y+16
66. Expand:
(3a−2b)(3a+2b)(3a - 2b)(3a + 2b)(3a−2b)(3a+2b)
Use the difference of squares formula (a−b)(a+b)=a2−b2(a - b)(a + b) = a^2 - b^2(a−b)
(a+b)=a2−b2:
(3a−2b)(3a+2b)=9a2−4b2(3a - 2b)(3a + 2b) = 9a^2 - 4b^2(3a−2b)(3a+2b)=9a2−4b2
67. Factor:
x2−9x^2 - 9x2−9
This is a difference of squares:
5. Solving the equation:
3z−14+z+23=5\{3z-1}{4} + \{z+2}{3} = 543z−1+3z+2=5
Find a common denominator (12):
3(3z−1)12+4(z+2)12=59z−3+4z+812=513z+512=513z+5=6013z=55z=5513≈4.23\{3(3z-1)}
{12} + \{4(z+2)}{12} = 5 \\ \{9z - 3 + 4z + 8}{12} = 5 \\ \{13z + 5}{12} = 5 \\ 13z + 5 =
60 \\ 13z = 55 \\ z = \{55}{13} \approx 4.23123(3z−1)+124(z+2)=5129z−3+4z+8=51213z+5
=513z+5=6013z=55z=1355≈4.23
6. Solving the equation:
2(a−3)+3(a+4)=52(a-3) + 3(a+4) = 52(a−3)+3(a+4)=5
Expand and combine like terms:
2a−6+3a+12=55a+6=55a=−1a=−152a - 6 + 3a + 12 = 5 \\ 5a + 6 = 5 \\ 5a = -1 \\ a = -\{1}
{5}2a−6+3a+12=55a+6=55a=−1a=−51
7. Solving the equation:
5(b+2)−2(b−3)=105(b+2) - 2(b-3) = 105(b+2)−2(b−3)=10
Expand and combine like terms:
5b+10−2b+6=103b+16=103b=−6b=−25b + 10 - 2b + 6 = 10 \\ 3b + 16 = 10 \\ 3b = -6 \\ b = -
25b+10−2b+6=103b+16=103b=−6b=−2
8. Solving the equation:
(x+1)2−4(x−2)=0(x+1)^2 - 4(x-2) = 0(x+1)2−4(x−2)=0
Expand and simplify:
(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0(x+1)^2 - 4x + 8 = 0 \\ x^2 + 2x + 1 - 4x + 8 =
0 \\ x^2 - 2x + 9 = 0(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0
Solve using the quadratic formula x=−b±b2−4ac2ax = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}x=2a−b±b2−4ac:
x=2±4−362x=2±−322x=1±i8x = \{2 \pm \sqrt{4 - 36}}{2} \\ x = \{2 \pm \sqrt{-32}}{2} \\ x
= 1 \pm i\sqrt{8}x=22±4−36x=22±−32x=1±i8
So, x=1+22ix = 1 + 2\sqrt{2}ix=1+22i and x=1−22ix = 1 - 2\sqrt{2}ix=1−22i.
9. Solving the equation:
3(y−1)2+2(y+3)=03(y-1)^2 + 2(y+3) = 03(y−1)2+2(y+3)=0
Expand and combine like terms:
3(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=03(y^2 - 2y + 1) + 2y + 6 = 0 \\ 3y^2 - 6y
+ 3 + 2y + 6 = 0 \\ 3y^2 - 4y + 9 = 03(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=0
Solve using the quadratic formula y=−b±b2−4ac2ay = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}y=2a−b±b2−4ac:
y=4±16−1086y=4±−926y=4±2i236y=2±i233y = \{4 \pm \sqrt{16 - 108}}{6} \\ y = \{4 \pm \
sqrt{-92}}{6} \\ y = \{4 \pm 2i\sqrt{23}}{6} \\ y = \{2 \pm i\sqrt{23}}{3}y=64±16−108
y=64±−92y=64±2i23y=32±i23
So, y=2+i233y = \{2 + i\sqrt{23}}{3}y=32+i23 and y=2−i233y = \{2 - i\sqrt{23}}
{3}y=32−i23.
10. Solving the equation:
(z−2)2−5(z+1)=0(z-2)^2 - 5(z+1) = 0(z−2)2−5(z+1)=0
Expand and simplify:
z2−4z+4−5z−5=0z2−9z−1=0z^2 - 4z + 4 - 5z - 5 = 0 \\ z^2 - 9z - 1 =
0z2−4z+4−5z−5=0z2−9z−1=0
Solve using the quadratic formula z=−b±b2−4ac2az = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}z=2a−b±b2−4ac:
z=9±81+42z=9±852z = \{9 \pm \sqrt{81 + 4}}{2} \\ z = \{9 \pm \sqrt{85}}{2}z=29±81+4
z=29±85
So, z=9+852z = \{9 + \sqrt{85}}{2}z=29+85 and z=9−852z = \{9 - \sqrt{85}}{2}z=29−85.
Advanced Algebraic Expressions
61. Simplify:
3x2−9x3x\frac{3x^2 - 9x}{3x}3x3x2−9x
Factor the numerator:
3x(x−3)3x=x−3\frac{3x(x - 3)}{3x} = x - 33x3x(x−3)=x−3
62. Simplify:
4y2−16y4y\frac{4y^2 - 16y}{4y}4y4y2−16y
Factor the numerator:
4y(y−4)4y=y−4\frac{4y(y - 4)}{4y} = y - 44y4y(y−4)=y−4
63. Simplify:
2z2+8z2z\frac{2z^2 + 8z}{2z}2z2z2+8z
Factor the numerator:
2z(z+4)2z=z+4\frac{2z(z + 4)}{2z} = z + 42z2z(z+4)=z+4
64. Expand:
(2x−3)2(2x - 3)^2(2x−3)2
Use the binomial expansion formula (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab +
b^2(a−b)2=a2−2ab+b2:
(2x−3)2=4x2−12x+9(2x - 3)^2 = 4x^2 - 12x + 9(2x−3)2=4x2−12x+9
65. Expand:
(y+4)2(y + 4)^2(y+4)2
Use the binomial expansion formula (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab +
b^2(a+b)2=a2+2ab+b2:
(y+4)2=y2+8y+16(y + 4)^2 = y^2 + 8y + 16(y+4)2=y2+8y+16
66. Expand:
(3a−2b)(3a+2b)(3a - 2b)(3a + 2b)(3a−2b)(3a+2b)
Use the difference of squares formula (a−b)(a+b)=a2−b2(a - b)(a + b) = a^2 - b^2(a−b)
(a+b)=a2−b2:
(3a−2b)(3a+2b)=9a2−4b2(3a - 2b)(3a + 2b) = 9a^2 - 4b^2(3a−2b)(3a+2b)=9a2−4b2
67. Factor:
x2−9x^2 - 9x2−9
This is a difference of squares:
5. Solving the equation:
3z−14+z+23=5\{3z-1}{4} + \{z+2}{3} = 543z−1+3z+2=5
Find a common denominator (12):
3(3z−1)12+4(z+2)12=59z−3+4z+812=513z+512=513z+5=6013z=55z=5513≈4.23\{3(3z-1)}
{12} + \{4(z+2)}{12} = 5 \\ \{9z - 3 + 4z + 8}{12} = 5 \\ \{13z + 5}{12} = 5 \\ 13z + 5 =
60 \\ 13z = 55 \\ z = \{55}{13} \approx 4.23123(3z−1)+124(z+2)=5129z−3+4z+8=51213z+5
=513z+5=6013z=55z=1355≈4.23
6. Solving the equation:
2(a−3)+3(a+4)=52(a-3) + 3(a+4) = 52(a−3)+3(a+4)=5
Expand and combine like terms:
2a−6+3a+12=55a+6=55a=−1a=−152a - 6 + 3a + 12 = 5 \\ 5a + 6 = 5 \\ 5a = -1 \\ a = -\{1}
{5}2a−6+3a+12=55a+6=55a=−1a=−51
7. Solving the equation:
5(b+2)−2(b−3)=105(b+2) - 2(b-3) = 105(b+2)−2(b−3)=10
Expand and combine like terms:
5b+10−2b+6=103b+16=103b=−6b=−25b + 10 - 2b + 6 = 10 \\ 3b + 16 = 10 \\ 3b = -6 \\ b = -
25b+10−2b+6=103b+16=103b=−6b=−2
8. Solving the equation:
(x+1)2−4(x−2)=0(x+1)^2 - 4(x-2) = 0(x+1)2−4(x−2)=0
Expand and simplify:
(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0(x+1)^2 - 4x + 8 = 0 \\ x^2 + 2x + 1 - 4x + 8 =
0 \\ x^2 - 2x + 9 = 0(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0
Solve using the quadratic formula x=−b±b2−4ac2ax = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}x=2a−b±b2−4ac:
x=2±4−362x=2±−322x=1±i8x = \{2 \pm \sqrt{4 - 36}}{2} \\ x = \{2 \pm \sqrt{-32}}{2} \\ x
= 1 \pm i\sqrt{8}x=22±4−36x=22±−32x=1±i8
So, x=1+22ix = 1 + 2\sqrt{2}ix=1+22i and x=1−22ix = 1 - 2\sqrt{2}ix=1−22i.
9. Solving the equation:
3(y−1)2+2(y+3)=03(y-1)^2 + 2(y+3) = 03(y−1)2+2(y+3)=0
Expand and combine like terms:
3(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=03(y^2 - 2y + 1) + 2y + 6 = 0 \\ 3y^2 - 6y
+ 3 + 2y + 6 = 0 \\ 3y^2 - 4y + 9 = 03(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=0
Solve using the quadratic formula y=−b±b2−4ac2ay = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}y=2a−b±b2−4ac:
y=4±16−1086y=4±−926y=4±2i236y=2±i233y = \{4 \pm \sqrt{16 - 108}}{6} \\ y = \{4 \pm \
sqrt{-92}}{6} \\ y = \{4 \pm 2i\sqrt{23}}{6} \\ y = \{2 \pm i\sqrt{23}}{3}y=64±16−108
y=64±−92y=64±2i23y=32±i23
So, y=2+i233y = \{2 + i\sqrt{23}}{3}y=32+i23 and y=2−i233y = \{2 - i\sqrt{23}}
{3}y=32−i23.
10. Solving the equation:
(z−2)2−5(z+1)=0(z-2)^2 - 5(z+1) = 0(z−2)2−5(z+1)=0
Expand and simplify:
z2−4z+4−5z−5=0z2−9z−1=0z^2 - 4z + 4 - 5z - 5 = 0 \\ z^2 - 9z - 1 =
0z2−4z+4−5z−5=0z2−9z−1=0
Solve using the quadratic formula z=−b±b2−4ac2az = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}z=2a−b±b2−4ac:
z=9±81+42z=9±852z = \{9 \pm \sqrt{81 + 4}}{2} \\ z = \{9 \pm \sqrt{85}}{2}z=29±81+4
z=29±85
So, z=9+852z = \{9 + \sqrt{85}}{2}z=29+85 and z=9−852z = \{9 - \sqrt{85}}{2}z=29−85.
Advanced Algebraic Expressions
61. Simplify:
3x2−9x3x\frac{3x^2 - 9x}{3x}3x3x2−9x
Factor the numerator:
3x(x−3)3x=x−3\frac{3x(x - 3)}{3x} = x - 33x3x(x−3)=x−3
62. Simplify:
4y2−16y4y\frac{4y^2 - 16y}{4y}4y4y2−16y
Factor the numerator:
4y(y−4)4y=y−4\frac{4y(y - 4)}{4y} = y - 44y4y(y−4)=y−4
63. Simplify:
2z2+8z2z\frac{2z^2 + 8z}{2z}2z2z2+8z
Factor the numerator:
2z(z+4)2z=z+4\frac{2z(z + 4)}{2z} = z + 42z2z(z+4)=z+4
64. Expand:
(2x−3)2(2x - 3)^2(2x−3)2
Use the binomial expansion formula (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab +
b^2(a−b)2=a2−2ab+b2:
(2x−3)2=4x2−12x+9(2x - 3)^2 = 4x^2 - 12x + 9(2x−3)2=4x2−12x+9
65. Expand:
(y+4)2(y + 4)^2(y+4)2
Use the binomial expansion formula (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab +
b^2(a+b)2=a2+2ab+b2:
(y+4)2=y2+8y+16(y + 4)^2 = y^2 + 8y + 16(y+4)2=y2+8y+16
66. Expand:
(3a−2b)(3a+2b)(3a - 2b)(3a + 2b)(3a−2b)(3a+2b)
Use the difference of squares formula (a−b)(a+b)=a2−b2(a - b)(a + b) = a^2 - b^2(a−b)
(a+b)=a2−b2:
(3a−2b)(3a+2b)=9a2−4b2(3a - 2b)(3a + 2b) = 9a^2 - 4b^2(3a−2b)(3a+2b)=9a2−4b2
67. Factor:
x2−9x^2 - 9x2−9
This is a difference of squares:
5. Solving the equation:
3z−14+z+23=5\{3z-1}{4} + \{z+2}{3} = 543z−1+3z+2=5
Find a common denominator (12):
3(3z−1)12+4(z+2)12=59z−3+4z+812=513z+512=513z+5=6013z=55z=5513≈4.23\{3(3z-1)}
{12} + \{4(z+2)}{12} = 5 \\ \{9z - 3 + 4z + 8}{12} = 5 \\ \{13z + 5}{12} = 5 \\ 13z + 5 =
60 \\ 13z = 55 \\ z = \{55}{13} \approx 4.23123(3z−1)+124(z+2)=5129z−3+4z+8=51213z+5
=513z+5=6013z=55z=1355≈4.23
6. Solving the equation:
2(a−3)+3(a+4)=52(a-3) + 3(a+4) = 52(a−3)+3(a+4)=5
Expand and combine like terms:
2a−6+3a+12=55a+6=55a=−1a=−152a - 6 + 3a + 12 = 5 \\ 5a + 6 = 5 \\ 5a = -1 \\ a = -\{1}
{5}2a−6+3a+12=55a+6=55a=−1a=−51
7. Solving the equation:
5(b+2)−2(b−3)=105(b+2) - 2(b-3) = 105(b+2)−2(b−3)=10
Expand and combine like terms:
5b+10−2b+6=103b+16=103b=−6b=−25b + 10 - 2b + 6 = 10 \\ 3b + 16 = 10 \\ 3b = -6 \\ b = -
25b+10−2b+6=103b+16=103b=−6b=−2
8. Solving the equation:
(x+1)2−4(x−2)=0(x+1)^2 - 4(x-2) = 0(x+1)2−4(x−2)=0
Expand and simplify:
(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0(x+1)^2 - 4x + 8 = 0 \\ x^2 + 2x + 1 - 4x + 8 =
0 \\ x^2 - 2x + 9 = 0(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0
Solve using the quadratic formula x=−b±b2−4ac2ax = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}x=2a−b±b2−4ac:
x=2±4−362x=2±−322x=1±i8x = \{2 \pm \sqrt{4 - 36}}{2} \\ x = \{2 \pm \sqrt{-32}}{2} \\ x
= 1 \pm i\sqrt{8}x=22±4−36x=22±−32x=1±i8
So, x=1+22ix = 1 + 2\sqrt{2}ix=1+22i and x=1−22ix = 1 - 2\sqrt{2}ix=1−22i.
9. Solving the equation:
3(y−1)2+2(y+3)=03(y-1)^2 + 2(y+3) = 03(y−1)2+2(y+3)=0
Expand and combine like terms:
3(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=03(y^2 - 2y + 1) + 2y + 6 = 0 \\ 3y^2 - 6y
+ 3 + 2y + 6 = 0 \\ 3y^2 - 4y + 9 = 03(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=0
Solve using the quadratic formula y=−b±b2−4ac2ay = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}y=2a−b±b2−4ac:
y=4±16−1086y=4±−926y=4±2i236y=2±i233y = \{4 \pm \sqrt{16 - 108}}{6} \\ y = \{4 \pm \
sqrt{-92}}{6} \\ y = \{4 \pm 2i\sqrt{23}}{6} \\ y = \{2 \pm i\sqrt{23}}{3}y=64±16−108
y=64±−92y=64±2i23y=32±i23
So, y=2+i233y = \{2 + i\sqrt{23}}{3}y=32+i23 and y=2−i233y = \{2 - i\sqrt{23}}
{3}y=32−i23.
10. Solving the equation:
(z−2)2−5(z+1)=0(z-2)^2 - 5(z+1) = 0(z−2)2−5(z+1)=0
Expand and simplify:
z2−4z+4−5z−5=0z2−9z−1=0z^2 - 4z + 4 - 5z - 5 = 0 \\ z^2 - 9z - 1 =
0z2−4z+4−5z−5=0z2−9z−1=0
Solve using the quadratic formula z=−b±b2−4ac2az = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}z=2a−b±b2−4ac:
z=9±81+42z=9±852z = \{9 \pm \sqrt{81 + 4}}{2} \\ z = \{9 \pm \sqrt{85}}{2}z=29±81+4
z=29±85
So, z=9+852z = \{9 + \sqrt{85}}{2}z=29+85 and z=9−852z = \{9 - \sqrt{85}}{2}z=29−85.
Advanced Algebraic Expressions
61. Simplify:
3x2−9x3x\frac{3x^2 - 9x}{3x}3x3x2−9x
Factor the numerator:
3x(x−3)3x=x−3\frac{3x(x - 3)}{3x} = x - 33x3x(x−3)=x−3
62. Simplify:
4y2−16y4y\frac{4y^2 - 16y}{4y}4y4y2−16y
Factor the numerator:
4y(y−4)4y=y−4\frac{4y(y - 4)}{4y} = y - 44y4y(y−4)=y−4
63. Simplify:
2z2+8z2z\frac{2z^2 + 8z}{2z}2z2z2+8z
Factor the numerator:
2z(z+4)2z=z+4\frac{2z(z + 4)}{2z} = z + 42z2z(z+4)=z+4
64. Expand:
(2x−3)2(2x - 3)^2(2x−3)2
Use the binomial expansion formula (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab +
b^2(a−b)2=a2−2ab+b2:
(2x−3)2=4x2−12x+9(2x - 3)^2 = 4x^2 - 12x + 9(2x−3)2=4x2−12x+9
65. Expand:
(y+4)2(y + 4)^2(y+4)2
Use the binomial expansion formula (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab +
b^2(a+b)2=a2+2ab+b2:
(y+4)2=y2+8y+16(y + 4)^2 = y^2 + 8y + 16(y+4)2=y2+8y+16
66. Expand:
(3a−2b)(3a+2b)(3a - 2b)(3a + 2b)(3a−2b)(3a+2b)
Use the difference of squares formula (a−b)(a+b)=a2−b2(a - b)(a + b) = a^2 - b^2(a−b)
(a+b)=a2−b2:
(3a−2b)(3a+2b)=9a2−4b2(3a - 2b)(3a + 2b) = 9a^2 - 4b^2(3a−2b)(3a+2b)=9a2−4b2
67. Factor:
x2−9x^2 - 9x2−9
This is a difference of squares:
5. Solving the equation:
3z−14+z+23=5\{3z-1}{4} + \{z+2}{3} = 543z−1+3z+2=5
Find a common denominator (12):
3(3z−1)12+4(z+2)12=59z−3+4z+812=513z+512=513z+5=6013z=55z=5513≈4.23\{3(3z-1)}
{12} + \{4(z+2)}{12} = 5 \\ \{9z - 3 + 4z + 8}{12} = 5 \\ \{13z + 5}{12} = 5 \\ 13z + 5 =
60 \\ 13z = 55 \\ z = \{55}{13} \approx 4.23123(3z−1)+124(z+2)=5129z−3+4z+8=51213z+5
=513z+5=6013z=55z=1355≈4.23
6. Solving the equation:
2(a−3)+3(a+4)=52(a-3) + 3(a+4) = 52(a−3)+3(a+4)=5
Expand and combine like terms:
2a−6+3a+12=55a+6=55a=−1a=−152a - 6 + 3a + 12 = 5 \\ 5a + 6 = 5 \\ 5a = -1 \\ a = -\{1}
{5}2a−6+3a+12=55a+6=55a=−1a=−51
7. Solving the equation:
5(b+2)−2(b−3)=105(b+2) - 2(b-3) = 105(b+2)−2(b−3)=10
Expand and combine like terms:
5b+10−2b+6=103b+16=103b=−6b=−25b + 10 - 2b + 6 = 10 \\ 3b + 16 = 10 \\ 3b = -6 \\ b = -
25b+10−2b+6=103b+16=103b=−6b=−2
8. Solving the equation:
(x+1)2−4(x−2)=0(x+1)^2 - 4(x-2) = 0(x+1)2−4(x−2)=0
Expand and simplify:
(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0(x+1)^2 - 4x + 8 = 0 \\ x^2 + 2x + 1 - 4x + 8 =
0 \\ x^2 - 2x + 9 = 0(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0
Solve using the quadratic formula x=−b±b2−4ac2ax = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}x=2a−b±b2−4ac:
x=2±4−362x=2±−322x=1±i8x = \{2 \pm \sqrt{4 - 36}}{2} \\ x = \{2 \pm \sqrt{-32}}{2} \\ x
= 1 \pm i\sqrt{8}x=22±4−36x=22±−32x=1±i8
So, x=1+22ix = 1 + 2\sqrt{2}ix=1+22i and x=1−22ix = 1 - 2\sqrt{2}ix=1−22i.
9. Solving the equation:
3(y−1)2+2(y+3)=03(y-1)^2 + 2(y+3) = 03(y−1)2+2(y+3)=0
Expand and combine like terms:
3(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=03(y^2 - 2y + 1) + 2y + 6 = 0 \\ 3y^2 - 6y
+ 3 + 2y + 6 = 0 \\ 3y^2 - 4y + 9 = 03(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=0
Solve using the quadratic formula y=−b±b2−4ac2ay = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}y=2a−b±b2−4ac:
y=4±16−1086y=4±−926y=4±2i236y=2±i233y = \{4 \pm \sqrt{16 - 108}}{6} \\ y = \{4 \pm \
sqrt{-92}}{6} \\ y = \{4 \pm 2i\sqrt{23}}{6} \\ y = \{2 \pm i\sqrt{23}}{3}y=64±16−108
y=64±−92y=64±2i23y=32±i23
So, y=2+i233y = \{2 + i\sqrt{23}}{3}y=32+i23 and y=2−i233y = \{2 - i\sqrt{23}}
{3}y=32−i23.
10. Solving the equation:
(z−2)2−5(z+1)=0(z-2)^2 - 5(z+1) = 0(z−2)2−5(z+1)=0
Expand and simplify:
z2−4z+4−5z−5=0z2−9z−1=0z^2 - 4z + 4 - 5z - 5 = 0 \\ z^2 - 9z - 1 =
0z2−4z+4−5z−5=0z2−9z−1=0
Solve using the quadratic formula z=−b±b2−4ac2az = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}z=2a−b±b2−4ac:
z=9±81+42z=9±852z = \{9 \pm \sqrt{81 + 4}}{2} \\ z = \{9 \pm \sqrt{85}}{2}z=29±81+4
z=29±85
So, z=9+852z = \{9 + \sqrt{85}}{2}z=29+85 and z=9−852z = \{9 - \sqrt{85}}{2}z=29−85.
Advanced Algebraic Expressions
61. Simplify:
3x2−9x3x\frac{3x^2 - 9x}{3x}3x3x2−9x
Factor the numerator:
3x(x−3)3x=x−3\frac{3x(x - 3)}{3x} = x - 33x3x(x−3)=x−3
62. Simplify:
4y2−16y4y\frac{4y^2 - 16y}{4y}4y4y2−16y
Factor the numerator:
4y(y−4)4y=y−4\frac{4y(y - 4)}{4y} = y - 44y4y(y−4)=y−4
63. Simplify:
2z2+8z2z\frac{2z^2 + 8z}{2z}2z2z2+8z
Factor the numerator:
2z(z+4)2z=z+4\frac{2z(z + 4)}{2z} = z + 42z2z(z+4)=z+4
64. Expand:
(2x−3)2(2x - 3)^2(2x−3)2
Use the binomial expansion formula (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab +
b^2(a−b)2=a2−2ab+b2:
(2x−3)2=4x2−12x+9(2x - 3)^2 = 4x^2 - 12x + 9(2x−3)2=4x2−12x+9
65. Expand:
(y+4)2(y + 4)^2(y+4)2
Use the binomial expansion formula (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab +
b^2(a+b)2=a2+2ab+b2:
(y+4)2=y2+8y+16(y + 4)^2 = y^2 + 8y + 16(y+4)2=y2+8y+16
66. Expand:
(3a−2b)(3a+2b)(3a - 2b)(3a + 2b)(3a−2b)(3a+2b)
Use the difference of squares formula (a−b)(a+b)=a2−b2(a - b)(a + b) = a^2 - b^2(a−b)
(a+b)=a2−b2:
(3a−2b)(3a+2b)=9a2−4b2(3a - 2b)(3a + 2b) = 9a^2 - 4b^2(3a−2b)(3a+2b)=9a2−4b2
67. Factor:
x2−9x^2 - 9x2−9
This is a difference of squares:
5. Solving the equation:
3z−14+z+23=5\{3z-1}{4} + \{z+2}{3} = 543z−1+3z+2=5
Find a common denominator (12):
3(3z−1)12+4(z+2)12=59z−3+4z+812=513z+512=513z+5=6013z=55z=5513≈4.23\{3(3z-1)}
{12} + \{4(z+2)}{12} = 5 \\ \{9z - 3 + 4z + 8}{12} = 5 \\ \{13z + 5}{12} = 5 \\ 13z + 5 =
60 \\ 13z = 55 \\ z = \{55}{13} \approx 4.23123(3z−1)+124(z+2)=5129z−3+4z+8=51213z+5
=513z+5=6013z=55z=1355≈4.23
6. Solving the equation:
2(a−3)+3(a+4)=52(a-3) + 3(a+4) = 52(a−3)+3(a+4)=5
Expand and combine like terms:
2a−6+3a+12=55a+6=55a=−1a=−152a - 6 + 3a + 12 = 5 \\ 5a + 6 = 5 \\ 5a = -1 \\ a = -\{1}
{5}2a−6+3a+12=55a+6=55a=−1a=−51
7. Solving the equation:
5(b+2)−2(b−3)=105(b+2) - 2(b-3) = 105(b+2)−2(b−3)=10
Expand and combine like terms:
5b+10−2b+6=103b+16=103b=−6b=−25b + 10 - 2b + 6 = 10 \\ 3b + 16 = 10 \\ 3b = -6 \\ b = -
25b+10−2b+6=103b+16=103b=−6b=−2
8. Solving the equation:
(x+1)2−4(x−2)=0(x+1)^2 - 4(x-2) = 0(x+1)2−4(x−2)=0
Expand and simplify:
(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0(x+1)^2 - 4x + 8 = 0 \\ x^2 + 2x + 1 - 4x + 8 =
0 \\ x^2 - 2x + 9 = 0(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0
Solve using the quadratic formula x=−b±b2−4ac2ax = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}x=2a−b±b2−4ac:
x=2±4−362x=2±−322x=1±i8x = \{2 \pm \sqrt{4 - 36}}{2} \\ x = \{2 \pm \sqrt{-32}}{2} \\ x
= 1 \pm i\sqrt{8}x=22±4−36x=22±−32x=1±i8
So, x=1+22ix = 1 + 2\sqrt{2}ix=1+22i and x=1−22ix = 1 - 2\sqrt{2}ix=1−22i.
9. Solving the equation:
3(y−1)2+2(y+3)=03(y-1)^2 + 2(y+3) = 03(y−1)2+2(y+3)=0
Expand and combine like terms:
3(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=03(y^2 - 2y + 1) + 2y + 6 = 0 \\ 3y^2 - 6y
+ 3 + 2y + 6 = 0 \\ 3y^2 - 4y + 9 = 03(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=0
Solve using the quadratic formula y=−b±b2−4ac2ay = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}y=2a−b±b2−4ac:
y=4±16−1086y=4±−926y=4±2i236y=2±i233y = \{4 \pm \sqrt{16 - 108}}{6} \\ y = \{4 \pm \
sqrt{-92}}{6} \\ y = \{4 \pm 2i\sqrt{23}}{6} \\ y = \{2 \pm i\sqrt{23}}{3}y=64±16−108
y=64±−92y=64±2i23y=32±i23
So, y=2+i233y = \{2 + i\sqrt{23}}{3}y=32+i23 and y=2−i233y = \{2 - i\sqrt{23}}
{3}y=32−i23.
10. Solving the equation:
(z−2)2−5(z+1)=0(z-2)^2 - 5(z+1) = 0(z−2)2−5(z+1)=0
Expand and simplify:
z2−4z+4−5z−5=0z2−9z−1=0z^2 - 4z + 4 - 5z - 5 = 0 \\ z^2 - 9z - 1 =
0z2−4z+4−5z−5=0z2−9z−1=0
Solve using the quadratic formula z=−b±b2−4ac2az = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}z=2a−b±b2−4ac:
z=9±81+42z=9±852z = \{9 \pm \sqrt{81 + 4}}{2} \\ z = \{9 \pm \sqrt{85}}{2}z=29±81+4
z=29±85
So, z=9+852z = \{9 + \sqrt{85}}{2}z=29+85 and z=9−852z = \{9 - \sqrt{85}}{2}z=29−85.
Advanced Algebraic Expressions
61. Simplify:
3x2−9x3x\frac{3x^2 - 9x}{3x}3x3x2−9x
Factor the numerator:
3x(x−3)3x=x−3\frac{3x(x - 3)}{3x} = x - 33x3x(x−3)=x−3
62. Simplify:
4y2−16y4y\frac{4y^2 - 16y}{4y}4y4y2−16y
Factor the numerator:
4y(y−4)4y=y−4\frac{4y(y - 4)}{4y} = y - 44y4y(y−4)=y−4
63. Simplify:
2z2+8z2z\frac{2z^2 + 8z}{2z}2z2z2+8z
Factor the numerator:
2z(z+4)2z=z+4\frac{2z(z + 4)}{2z} = z + 42z2z(z+4)=z+4
64. Expand:
(2x−3)2(2x - 3)^2(2x−3)2
Use the binomial expansion formula (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab +
b^2(a−b)2=a2−2ab+b2:
(2x−3)2=4x2−12x+9(2x - 3)^2 = 4x^2 - 12x + 9(2x−3)2=4x2−12x+9
65. Expand:
(y+4)2(y + 4)^2(y+4)2
Use the binomial expansion formula (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab +
b^2(a+b)2=a2+2ab+b2:
(y+4)2=y2+8y+16(y + 4)^2 = y^2 + 8y + 16(y+4)2=y2+8y+16
66. Expand:
(3a−2b)(3a+2b)(3a - 2b)(3a + 2b)(3a−2b)(3a+2b)
Use the difference of squares formula (a−b)(a+b)=a2−b2(a - b)(a + b) = a^2 - b^2(a−b)
(a+b)=a2−b2:
(3a−2b)(3a+2b)=9a2−4b2(3a - 2b)(3a + 2b) = 9a^2 - 4b^2(3a−2b)(3a+2b)=9a2−4b2
67. Factor:
x2−9x^2 - 9x2−9
This is a difference of squares:
5. Solving the equation:
3z−14+z+23=5\{3z-1}{4} + \{z+2}{3} = 543z−1+3z+2=5
Find a common denominator (12):
3(3z−1)12+4(z+2)12=59z−3+4z+812=513z+512=513z+5=6013z=55z=5513≈4.23\{3(3z-1)}
{12} + \{4(z+2)}{12} = 5 \\ \{9z - 3 + 4z + 8}{12} = 5 \\ \{13z + 5}{12} = 5 \\ 13z + 5 =
60 \\ 13z = 55 \\ z = \{55}{13} \approx 4.23123(3z−1)+124(z+2)=5129z−3+4z+8=51213z+5
=513z+5=6013z=55z=1355≈4.23
6. Solving the equation:
2(a−3)+3(a+4)=52(a-3) + 3(a+4) = 52(a−3)+3(a+4)=5
Expand and combine like terms:
2a−6+3a+12=55a+6=55a=−1a=−152a - 6 + 3a + 12 = 5 \\ 5a + 6 = 5 \\ 5a = -1 \\ a = -\{1}
{5}2a−6+3a+12=55a+6=55a=−1a=−51
7. Solving the equation:
5(b+2)−2(b−3)=105(b+2) - 2(b-3) = 105(b+2)−2(b−3)=10
Expand and combine like terms:
5b+10−2b+6=103b+16=103b=−6b=−25b + 10 - 2b + 6 = 10 \\ 3b + 16 = 10 \\ 3b = -6 \\ b = -
25b+10−2b+6=103b+16=103b=−6b=−2
8. Solving the equation:
(x+1)2−4(x−2)=0(x+1)^2 - 4(x-2) = 0(x+1)2−4(x−2)=0
Expand and simplify:
(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0(x+1)^2 - 4x + 8 = 0 \\ x^2 + 2x + 1 - 4x + 8 =
0 \\ x^2 - 2x + 9 = 0(x+1)2−4x+8=0x2+2x+1−4x+8=0x2−2x+9=0
Solve using the quadratic formula x=−b±b2−4ac2ax = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}x=2a−b±b2−4ac:
x=2±4−362x=2±−322x=1±i8x = \{2 \pm \sqrt{4 - 36}}{2} \\ x = \{2 \pm \sqrt{-32}}{2} \\ x
= 1 \pm i\sqrt{8}x=22±4−36x=22±−32x=1±i8
So, x=1+22ix = 1 + 2\sqrt{2}ix=1+22i and x=1−22ix = 1 - 2\sqrt{2}ix=1−22i.
9. Solving the equation:
3(y−1)2+2(y+3)=03(y-1)^2 + 2(y+3) = 03(y−1)2+2(y+3)=0
Expand and combine like terms:
3(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=03(y^2 - 2y + 1) + 2y + 6 = 0 \\ 3y^2 - 6y
+ 3 + 2y + 6 = 0 \\ 3y^2 - 4y + 9 = 03(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=0
Solve using the quadratic formula y=−b±b2−4ac2ay = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}y=2a−b±b2−4ac:
y=4±16−1086y=4±−926y=4±2i236y=2±i233y = \{4 \pm \sqrt{16 - 108}}{6} \\ y = \{4 \pm \
sqrt{-92}}{6} \\ y = \{4 \pm 2i\sqrt{23}}{6} \\ y = \{2 \pm i\sqrt{23}}{3}y=64±16−108
y=64±−92y=64±2i23y=32±i23
So, y=2+i233y = \{2 + i\sqrt{23}}{3}y=32+i23 and y=2−i233y = \{2 - i\sqrt{23}}
{3}y=32−i23.
10. Solving the equation:
(z−2)2−5(z+1)=0(z-2)^2 - 5(z+1) = 0(z−2)2−5(z+1)=0
Expand and simplify:
z2−4z+4−5z−5=0z2−9z−1=0z^2 - 4z + 4 - 5z - 5 = 0 \\ z^2 - 9z - 1 =
0z2−4z+4−5z−5=0z2−9z−1=0
Solve using the quadratic formula z=−b±b2−4ac2az = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}z=2a−b±b2−4ac:
z=9±81+42z=9±852z = \{9 \pm \sqrt{81 + 4}}{2} \\ z = \{9 \pm \sqrt{85}}{2}z=29±81+4
z=29±85
So, z=9+852z = \{9 + \sqrt{85}}{2}z=29+85 and z=9−852z = \{9 - \sqrt{85}}{2}z=29−85.
Advanced Algebraic Expressions
61. Simplify:
3x2−9x3x\frac{3x^2 - 9x}{3x}3x3x2−9x
Factor the numerator:
3x(x−3)3x=x−3\frac{3x(x - 3)}{3x} = x - 33x3x(x−3)=x−3
62. Simplify:
4y2−16y4y\frac{4y^2 - 16y}{4y}4y4y2−16y
Factor the numerator:
4y(y−4)4y=y−4\frac{4y(y - 4)}{4y} = y - 44y4y(y−4)=y−4
63. Simplify:
2z2+8z2z\frac{2z^2 + 8z}{2z}2z2z2+8z
Factor the numerator:
2z(z+4)2z=z+4\frac{2z(z + 4)}{2z} = z + 42z2z(z+4)=z+4
64. Expand:
(2x−3)2(2x - 3)^2(2x−3)2
Use the binomial expansion formula (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab +
b^2(a−b)2=a2−2ab+b2:
(2x−3)2=4x2−12x+9(2x - 3)^2 = 4x^2 - 12x + 9(2x−3)2=4x2−12x+9
65. Expand:
(y+4)2(y + 4)^2(y+4)2
Use the binomial expansion formula (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab +
b^2(a+b)2=a2+2ab+b2:
(y+4)2=y2+8y+16(y + 4)^2 = y^2 + 8y + 16(y+4)2=y2+8y+16
66. Expand:
(3a−2b)(3a+2b)(3a - 2b)(3a + 2b)(3a−2b)(3a+2b)
Use the difference of squares formula (a−b)(a+b)=a2−b2(a - b)(a + b) = a^2 - b^2(a−b)
(a+b)=a2−b2:
(3a−2b)(3a+2b)=9a2−4b2(3a - 2b)(3a + 2b) = 9a^2 - 4b^2(3a−2b)(3a+2b)=9a2−4b2
67. Factor:
x2−9x^2 - 9x2−9
This is a difference of squares:
9. Solving the equation:
3(y−1)2+2(y+3)=03(y-1)^2 + 2(y+3) = 03(y−1)2+2(y+3)=0
Expand and combine like terms:
3(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=03(y^2 - 2y + 1) + 2y + 6 = 0 \\ 3y^2 - 6y
+ 3 + 2y + 6 = 0 \\ 3y^2 - 4y + 9 = 03(y2−2y+1)+2y+6=03y2−6y+3+2y+6=03y2−4y+9=0
Solve using the quadratic formula y=−b±b2−4ac2ay = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}y=2a−b±b2−4ac:
y=4±16−1086y=4±−926y=4±2i236y=2±i233y = \{4 \pm \sqrt{16 - 108}}{6} \\ y = \{4 \pm \
sqrt{-92}}{6} \\ y = \{4 \pm 2i\sqrt{23}}{6} \\ y = \{2 \pm i\sqrt{23}}{3}y=64±16−108
y=64±−92y=64±2i23y=32±i23
So, y=2+i233y = \{2 + i\sqrt{23}}{3}y=32+i23 and y=2−i233y = \{2 - i\sqrt{23}}
{3}y=32−i23.
10. Solving the equation:
(z−2)2−5(z+1)=0(z-2)^2 - 5(z+1) = 0(z−2)2−5(z+1)=0
Expand and simplify:
z2−4z+4−5z−5=0z2−9z−1=0z^2 - 4z + 4 - 5z - 5 = 0 \\ z^2 - 9z - 1 =
0z2−4z+4−5z−5=0z2−9z−1=0
Solve using the quadratic formula z=−b±b2−4ac2az = \{-b \pm \sqrt{b^2 - 4ac}}
{2a}z=2a−b±b2−4ac:
z=9±81+42z=9±852z = \{9 \pm \sqrt{81 + 4}}{2} \\ z = \{9 \pm \sqrt{85}}{2}z=29±81+4
z=29±85
So, z=9+852z = \{9 + \sqrt{85}}{2}z=29+85 and z=9−852z = \{9 - \sqrt{85}}{2}z=29−85.
Advanced Algebraic Expressions
61. Simplify:
3x2−9x3x\frac{3x^2 - 9x}{3x}3x3x2−9x
Factor the numerator:
3x(x−3)3x=x−3\frac{3x(x - 3)}{3x} = x - 33x3x(x−3)=x−3
62. Simplify:
4y2−16y4y\frac{4y^2 - 16y}{4y}4y4y2−16y
Factor the numerator:
4y(y−4)4y=y−4\frac{4y(y - 4)}{4y} = y - 44y4y(y−4)=y−4
63. Simplify:
2z2+8z2z\frac{2z^2 + 8z}{2z}2z2z2+8z
Factor the numerator:
2z(z+4)2z=z+4\frac{2z(z + 4)}{2z} = z + 42z2z(z+4)=z+4
64. Expand:
(2x−3)2(2x - 3)^2(2x−3)2
Use the binomial expansion formula (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab +
b^2(a−b)2=a2−2ab+b2:
(2x−3)2=4x2−12x+9(2x - 3)^2 = 4x^2 - 12x + 9(2x−3)2=4x2−12x+9
65. Expand:
(y+4)2(y + 4)^2(y+4)2
Use the binomial expansion formula (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab +
b^2(a+b)2=a2+2ab+b2:
(y+4)2=y2+8y+16(y + 4)^2 = y^2 + 8y + 16(y+4)2=y2+8y+16
66. Expand:
(3a−2b)(3a+2b)(3a - 2b)(3a + 2b)(3a−2b)(3a+2b)
Use the difference of squares formula (a−b)(a+b)=a2−b2(a - b)(a + b) = a^2 - b^2(a−b)
(a+b)=a2−b2:
(3a−2b)(3a+2b)=9a2−4b2(3a - 2b)(3a + 2b) = 9a^2 - 4b^2(3a−2b)(3a+2b)=9a2−4b2
67. Factor:
x2−9x^2 - 9x2−9
This is a difference of squares:
x2−9=(x−3)(x+3)x^2 - 9 = (x - 3)(x + 3)x2−9=(x−3)(x+3)
68. Factor:
4y2−14y^2 - 14y2−1
This is a difference of squares:
4y2−1=(2y−1)(2y+1)4y^2 - 1 = (2y - 1)(2y + 1)4y2−1=(2y−1)(2y+1)
69. Factor:
9z2−259z^2 - 259z2−25
This is a difference of squares:
9z2−25=(3z−5)(3z+5)9z^2 - 25 = (3z - 5)(3z + 5)9z2−25=(3z−5)(3z+5)
70. Factor:
16a2−116a^2 - 116a2−1
This is a difference of squares:
16a2−1=(4a−1)(4a+1)16a^2 - 1 = (4a - 1)(4a + 1)16a2−1=(4a−1)(4a+1)
Rational Expressions
71. Simplify:
x2−1x+1\frac{x^2 - 1}{x + 1}x+1x2−1
Factor the numerator:
(x−1)(x+1)x+1=x−1(for5x≠−1)\frac{(x - 1)(x + 1)}{x + 1} = x - 1 \quad \text{(for } x \neq -1\
text{)}x+1(x−1)(x+1)=x−1(for5x =−1)
72. Simplify:
y2−4y−2\frac{y^2 - 4}{y - 2}y−2y2−4
Factor the numerator:
(y−2)(y+2)y−2=y+2(for5y≠2)\frac{(y - 2)(y + 2)}{y - 2} = y + 2 \quad \text{(for } y \neq 2\
text{)}y−2(y−2)(y+2)=y+2(for5y =2)
73. Simplify:
z2−9z+3\frac{z^2 - 9}{z + 3}z+3z2−9
Factor the numerator:
(z−3)(z+3)z+3=z−3(for5z≠−3)\frac{(z - 3)(z + 3)}{z + 3} = z - 3 \quad \text{(for } z \neq -3\
text{)}z+3(z−3)(z+3)=z−3(for5z =−3)
74. Simplify:
4a2−162a+4\frac{4a^2 - 16}{2a + 4}2a+44a2−16
Factor both numerator and denominator:
4(a2−4)2(a+2)=4(a−2)(a+2)2(a+2)=2(a−2)(for5a≠−2)\frac{4(a^2 - 4)}{2(a + 2)} = \frac{4(a -
2)(a + 2)}{2(a + 2)} = 2(a - 2) \quad \text{(for } a \neq -2\text{)}2(a+2)4(a2−4)
=2(a+2)4(a−2)(a+2)=2(a−2)(for5a =−2)
75. Simplify:
9b2−253b−5\frac{9b^2 - 25}{3b - 5}3b−59b2−25
Factor the numerator:
(3b−5)(3b+5)3b−5=3b+5(for5b≠53)\frac{(3b - 5)(3b + 5)}{3b - 5} = 3b + 5 \quad \text{(for }
b \neq \frac{5}{3}\text{)}3b−5(3b−5)(3b+5)=3b+5(for5b =35)
76. Solve for xxx:
3x+2x−1=4\frac{3x + 2}{x - 1} = 4x−13x+2=4
Multiply both sides by x−1x - 1x−1:
3x+2=4(x−1)3x+2=4x−42+4=4x−3xx=63x + 2 = 4(x - 1) \\ 3x + 2 = 4x - 4 \\ 2 + 4 = 4x -
3x \\ x = 63x+2=4(x−1)3x+2=4x−42+4=4x−3xx=6
77. Solve for yyy:
2y+1y+3=3\frac{2y + 1}{y + 3} = 3y+32y+1=3
Multiply both sides by y+3y + 3y+3:
2y+1=3(y+3)2y+1=3y+91−9=3y−2yy=−82y + 1 = 3(y + 3) \\ 2y + 1 = 3y + 9 \\ 1 - 9 = 3y -
2y \\ y = -82y+1=3(y+3)2y+1=3y+91−9=3y−2yy=−8
78. Solve for zzz:
5z−3z+2=2\frac{5z - 3}{z + 2} = 2z+25z−3=2
Multiply both sides by z+2z + 2z+2:
5z−3=2(z+2)5z−3=2z+45z−2z=4+33z=7z=735z - 3 = 2(z + 2) \\ 5z - 3 = 2z + 4 \\ 5z - 2z = 4
+ 3 \\ 3z = 7 \\ z = \frac{7}{3}5z−3=2(z+2)5z−3=2z+45z−2z=4+33z=7z=37
79. Solve for aaa:
4a+52a−3=1\frac{4a + 5}{2a - 3} = 12a−34a+5=1
Multiply both sides by 2a−32a - 32a−3:
4a+5=2a−34a−2a=−3−52a=−8a=−44a + 5 = 2a - 3 \\ 4a - 2a = -3 - 5 \\ 2a = -8 \\ a = -
44a+5=2a−34a−2a=−3−52a=−8a=−4
80. Solve for bbb:
6b−7b+4=5\frac{6b - 7}{b + 4} = 5b+46b−7=5
Multiply both sides by b+4b + 4b+4:
6b−7=5(b+4)6b−7=5b+206b−5b=20+7b=276b - 7 = 5(b + 4) \\ 6b - 7 = 5b + 20 \\ 6b - 5b =
20 + 7 \\ b = 276b−7=5(b+4)6b−7=5b+206b−5b=20+7b=27
Exponential and Logarithmic Functions
81. Solve for xxx:
2x=162^x = 162x=16
Rewrite 16 as 242^424:
2x=24x=42^x = 2^4 \\ x = 42x=24x=4
82. Solve for yyy:
3y=273^y = 273y=27
Rewrite 27 as 333^333:
3y=33y=33^y = 3^3 \\ y = 33y=33y=3
83. Solve for zzz:
5z=1255^z = 1255z=125
Rewrite 125 as 535^353:
5z=53z=35^z = 5^3 \\ z = 35z=53z=3
84. Solve for aaa:
2a+1=322^{a+1} = 322a+1=32
Rewrite 32 as 252^525:
2a+1=25a+1=5a=42^{a+1} = 2^5 \\ a + 1 = 5 \\ a = 42a+1=25a+1=5a=4
85. Solve for bbb:
3b−2=13^{b-2} = 13b−2=1
Rewrite 1 as 303^030:
3b−2=30b−2=0b=23^{b-2} = 3^0 \\ b - 2 = 0 \\ b = 23b−2=30b−2=0b=2
86. Solve for xxx:
log 2x=3\log_2 x = 3log2x=3
Rewrite in exponential form:
x=23x=8x = 2^3 \\ x = 8x=23x=8
87. Solve for yyy:
log 3y=2\log_3 y = 2log3y=2
Rewrite in exponential form:
y=32y=9y = 3^2 \\ y = 9y=32y=9
88. Solve for zzz:
log 5z=3\log_5 z = 3log5z=3
Rewrite in exponential form:
z=53z=125z = 5^3 \\ z = 125z=53z=125
89. Solve for aaa:
log 4a=2\log_4 a = 2log4a=2
Rewrite in exponential form:
a=42a=16a = 4^2 \\ a = 16a=42a=16
90. Solve for bbb:
log 6b=1\log_6 b = 1log6b=1
Rewrite in exponential form:
b=61b=6b = 6^1 \\ b = 6b=61b=6
Problem 1:
Evaluate the integral:
∫01ln (1+x2)1+x2 dx\int_0^1 \frac{\ln(1 + x^2)}{1 + x^2} \, dx∫011+x2ln(1+x2)dx
Problem 2:
Evaluate the integral:
∫0πxsin (x) dx\int_0^\pi x \sin(x) \, dx∫0πxsin(x)dx
Solutions:
Solution to Problem 1:
To evaluate the integral
I=∫01ln (1+x2)1+x2 dx,I = \int_0^1 \frac{\ln(1 + x^2)}{1 + x^2} \, dx,I=∫011+x2ln(1+x2)dx,
we use the substitution x=tan (θ)x = \tan(\theta)x=tan(θ). This gives us dx=sec 2(θ) dθdx = \
sec^2(\theta) \, d\thetadx=sec2(θ)dθ, and the limits change from x=0x = 0x=0 to x=1x = 1x=1
correspond to θ=0\theta = 0θ=0 to θ=π4\theta = \frac{\pi}{4}θ=4π.
Thus, the integral becomes:
I=∫0π4ln (1+tan 2(θ))sec 2(θ)1+tan 2(θ) dθ.I = \int_0^{\frac{\pi}{4}} \frac{\ln(1 + \tan^2(\
theta)) \sec^2(\theta)}{1 + \tan^2(\theta)} \, d\theta.I=∫04π1+tan2(θ)ln(1+tan2(θ))sec2(θ)dθ.
Since 1+tan 2(θ)=sec 2(θ)1 + \tan^2(\theta) = \sec^2(\theta)1+tan2(θ)=sec2(θ), the integrand
simplifies to:
I=∫0π4ln (sec 2(θ)) dθ=2∫0π4ln (sec (θ)) dθ.I = \int_0^{\frac{\pi}{4}} \ln(\sec^2(\theta)) \, d\
theta = 2 \int_0^{\frac{\pi}{4}} \ln(\sec(\theta)) \, d\theta.I=∫04πln(sec2(θ))dθ=2∫04π
ln(sec(θ))dθ.
Using the identity sec (θ)=1cos (θ)\sec(\theta) = \frac{1}{\cos(\theta)}sec(θ)=cos(θ)1, we get:
I=2∫0π4ln (1cos (θ)) dθ=2∫0π4(−ln (cos (θ))) dθ.I = 2 \int_0^{\frac{\pi}{4}} \ln\left(\frac{1}{\
cos(\theta)}\right) \, d\theta = 2 \int_0^{\frac{\pi}{4}} (-\ln(\cos(\theta))) \, d\theta.I=2∫04π
ln(cos(θ)1)dθ=2∫04π(−ln(cos(θ)))dθ.
Thus,
I=−2∫0π4ln (cos (θ)) dθ.I = -2 \int_0^{\frac{\pi}{4}} \ln(\cos(\theta)) \, d\theta.I=−2∫04π
ln(cos(θ))dθ.
It is known that
∫0π4ln (cos (θ)) dθ=−π4ln 2.\int_0^{\frac{\pi}{4}} \ln(\cos(\theta)) \, d\theta = -\frac{\pi}{4} \
ln 2.∫04πln(cos(θ))dθ=−4πln2.
So,
I=−2(−π4ln 2)=π2ln 2.I = -2 \left( -\frac{\pi}{4} \ln 2 \right) = \frac{\pi}{2} \ln 2.I=−2(−4π
ln2)=2πln2.
Therefore, the integral evaluates to:
π2ln 2\boxed{\frac{\pi}{2} \ln 2}2πln2
Solution to Problem 2:
To evaluate the integral
I=∫0πxsin (x) dx,I = \int_0^\pi x \sin(x) \, dx,I=∫0πxsin(x)dx,
we use integration by parts. Let u=xu = xu=x and dv=sin (x) dxdv = \sin(x) \, dxdv=sin(x)dx.
Then du=dxdu = dxdu=dx and v=−cos (x)v = -\cos(x)v=−cos(x).
Using integration by parts, ∫u dv=uv−∫v du\int u \, dv = uv - \int v \, du∫udv=uv−∫vdu, we get:
I=−xcos (x)∣0π+∫0πcos (x) dx.I = \left. -x \cos(x) \right|_0^\pi + \int_0^\pi \cos(x) \,
dx.I=−xcos(x)∣0π+∫0πcos(x)dx.
Evaluating the boundary term:
−xcos (x)∣0π=−πcos (π)+0⋅cos (0)=−π(−1)+0=π.-x \cos(x) \bigg|_0^\pi = -\pi \cos(\pi) + 0 \
cdot \cos(0) = -\pi(-1) + 0 = \pi.−xcos(x)0π=−πcos(π)+0⋅cos(0)=−π(−1)+0=π.
Next, evaluate the remaining integral:
∫0πcos (x) dx=sin (x)∣0π=sin (π)−sin (0)=0−0=0.\int_0^\pi \cos(x) \, dx = \sin(x) \bigg|_0^\pi = \
sin(\pi) - \sin(0) = 0 - 0 = 0.∫0πcos(x)dx=sin(x)0π=sin(π)−sin(0)=0−0=0.
Combining these results, we get:
I=π+0=π.I = \pi + 0 = \pi.I=π+0=π.
Therefore, the integral evaluates to:
π\boxed{\pi}π
