Algebraic Summary

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Algebra-QuadraticEquations.pptx

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GCSE Mathematics – Quadratic Equations

SLIDE NUMBER 1

May 2019

Rebecca Wigfull

Insert a relevant picture here to fill the whole space

Session Objectives

The purpose of the session is to be able to:

Recall the key characteristics of linear and quadratic graphs.

Factorise a quadratic equation including the difference of two squares.

Apply factorising to solving a quadratic equation.

Solve quadratic equations by completing the square.

Solve quadratic equations by using the quadratic formula.

Solve quadratic inequalities.

Solve quadratic equations using an iterative method.

SLIDE NUMBER 2

May 2019

CONSIDER…

At this point you should consider the list of session objectives and ask yourself:

How many of the session objectives am I confident with

Could I explain these objectives in relation to teaching and learning

SLIDE NUMBER 3

May 2019

Session Objectives

SLIDE NUMBER 4

May 2019

The purpose of the session is to be able to:

Recall the key characteristics of linear and quadratic graphs.

Factorise a quadratic equation including the difference of two squares.

Apply factorising to solving a quadratic equation.

Solve quadratic equations by completing the square.

Solve quadratic equations by using the quadratic formula.

Solve quadratic inequalities.

Solve quadratic equations using an iterative method.

Linear Graphs

Quadratic Equations

SLIDE NUMBER 5

May 2019

y = mx + c

Negative curve

Positive curve

Negative curve

Positive curve

Quadratic Graphs

Quadratic Equations

SLIDE NUMBER 6

May 2019

y = ax2 +bx + c

Key Characteristics of a Quadratic Graph

Quadratic Equations

SLIDE NUMBER 7

May 2019

Root

(-2,0)

(2,0)

Turning point

(0, -4)

Positive curve

Key Characteristics of a Quadratic Graph

Quadratic Equations

SLIDE NUMBER 8

May 2019

Root

(-4,0)

(2,0)

Turning point

(-1, 9)

Negative curve

Y - intercept

(0, 8)

Key Characteristics of a Quadratic Graph

Quadratic Equations

SLIDE NUMBER 9

May 2019

Root

Turning point

(-1, 0)

Positive curve

y - intercept

(0, 1)

Repeated root

Key Characteristics of a Quadratic Graph

Quadratic Equations

SLIDE NUMBER 10

May 2019

Turning point

(3, 2)

Positive curve

y - intercept

(0, 11)

Turning point

(-1, -1)

Negative curve

(0, -2)

Y - intercept

No Roots

Review of main ideas from above:

For the following 3 graphs list the key characteristic of each one.

SLIDE NUMBER 11

May 2019

CONSIDER…

The green graph is a positive quadratic curve.

It has no roots.

The turning point is (1,2).

The intercept on the y-axis is (0,3).

The blue graph is a positive quadratic curve.

It’s roots are at (-3,0) and (3,0).

The turning point is (0,-9).

The intercept on the y-axis is (0,-9).

The red graph is a negative quadratic curve.

It has a repeated root where x = 2.

The turning point is (2,0).

The intercept on the y-axis is (0, -4).

Session Objectives

SLIDE NUMBER 12

May 2019

The purpose of the session is to be able to:

Recall the key characteristics of linear and quadratic graphs.

Factorise a quadratic equation including the difference of two squares.

Apply factorising to solving a quadratic equation.

Solve quadratic equations by completing the square.

Solve quadratic equations by using the quadratic formula.

Solve quadratic inequalities.

Solve quadratic equations using an iterative method.

Factorising Quadratics

Quadratic Equations

SLIDE NUMBER 13

May 2019

If (x + a)(x + b) is expanded we get

(x + a)(x + b) = x2 + ax + bx + ab

= x2 + (a + b)x + ab

Factorising Quadratics

Quadratic Equations

SLIDE NUMBER 14

May 2019

Quadratic	Factorised	Rule applied
x2 + 5x + 6	(x + 3)(x + 2)	A positive number in each bracket.
x2 - 5x + 6	(x - 3)(x - 2)	A negative number in each bracket.
x2 - x - 6	(x - 3)(x + 2)	A negative and a positive number in each bracket.
x2 + x - 6	(x + 3)(x - 2)	A negative and a positive number in each bracket.

Factorising Quadratics

Quadratic Equations

SLIDE NUMBER 15

May 2019

x2 + 6x + 8

(x )(x )

(x + )(x + )

x2 + 6x + 8

a	b	total
1	8	1+8 = 9
2	4	2+4 = 6

(x + a )(x + b )

(x + 2 )(x + 4 )

= x2 + 4x + 2x + 8

= x2 + 6x + 8

Factorising Quadratics

Quadratic Equations

SLIDE NUMBER 16

May 2019

x2 - 7x + 12

(x )(x )

(x - )(x - )

x2 - 7x + 12

a	b	total
1	12	1+12 = 13
2	6	2+6 = 8
3	4	3+4=7

(x - a )(x - b )

(x - 3 )(x - 4 )

= x2 - 4x - 3x + 12

= x2 - 7x + 12

Factorising Quadratics

Quadratic Equations

SLIDE NUMBER 17

May 2019

x2 + 6x - 16

(x )(x )

(x + )(x - )

a	b	difference
1	16	15
2	8	6
4	4	0

(x + 8 )(x - 2 )

= x2 - 2x + 8x - 16

= x2 + 6x - 16

Factorising Quadratics

Quadratic Equations

SLIDE NUMBER 18

May 2019

x2 - x - 20

(x )(x )

(x + )(x - )

a	b	difference
1	20	19
2	10	8
4	5	1

(x + 4 )(x - 5 )

(x + 4)(x - 5 )

= x2 - 5x + 4x - 20

= x2 - x - 20

Factorising Quadratics

Quadratic Equations

SLIDE NUMBER 19

May 2019

x2 - 36

(x )(x )

(x + )(x - )

(x + 6 )(x - 6 )

(x + 6)(x - 6 )

= x2 - 6x + 6x - 36

= x2 - 36

The difference of two squares

Factorising Quadratics

Quadratic Equations

SLIDE NUMBER 20

May 2019

9x2 - 36

( )( )

( + )( - )

(3x + 6 )(3x - 6 )

(3x + 6)(3x - 6 )

= 9x2 - 18x + 18x - 36

= 9x2 - 36

The difference of two squares

Factorising Quadratics – single bracket

Quadratic Equations

SLIDE NUMBER 21

May 2019

x2 - 23x

x(x )

= x2 - 23x

x(x - 23 )

2x2 + 14x

2x(x )

= 2x2 + 14x

2x(x + 7 )

Factorising Quadratics

– where the coefficient of x2 is not 1

Quadratic Equations

SLIDE NUMBER 22

May 2019

Can you do an initial factorisation to make the coefficient of x2 1?

5x2 - 20

5(x2 - 4)

5(x + 2)(x - 2)

Difference of two squares.

Factorising Quadratics

– where the coefficient of x2 is not 1 (prime)

Quadratic Equations

SLIDE NUMBER 23

May 2019

3x2 + 11x + 6

(3x + )(x + )

	3x	Total
1	6	19
2	3	11
3	2	9
6	1	9

(3x + 2 )(x + 3 )

Factorising Quadratics

– where the coefficient of x2 is not 1 (prime)

- alternative method

Quadratic Equations

SLIDE NUMBER 24

May 2019

3x2 + 11x + 6

(3x + )(x + )

3 x 6 = 18

a	b	total
1	18	19
2	9	11
3	6	9

(3x + 2)(x + 3)

Factorising Quadratics

- where the coefficient of x2 is not 1 (prime)

Quadratic Equations

SLIDE NUMBER 25

May 2019

5x2 + 7x - 6

(5x )(x )

a	b
1	-6	-29
2	-3	-13
3	-2	-7
6	-1	1

(5x - 3)(x + 2)

a	b
-1	6	29
-2	3	13
-3	2	7
-6	1	-1

Factorising Quadratics

- where the coefficient of x2 is not 1 (prime)

Quadratic Equations

SLIDE NUMBER 26

May 2019

5x2 + 7x - 6

(5x )(x )

5 x 6 = 30

(5x - 3)(x + 2)

a	b	Difference
1	30	29
2	15	13
3	10	7
6	5	1

Factorising Quadratics

- where the coefficient of x2 is not 1 (not a prime)

Quadratic Equations

SLIDE NUMBER 27

May 2019

8x2 - 2x - 15

(2x )(4x )

a	b
1	-15	-119
3	-5	-37
5	-3	-19
15	-1	7

a	b
-1	15	119
-3	5	37
-5	3	19
-15	1	-7

(8x )(x )

Factorising Quadratics

- where the coefficient of x2 is not 1 ( not a prime)

Quadratic Equations

SLIDE NUMBER 28

May 2019

8x2 - 2x - 15

(2x )(4x )

a	b
1	-15	-26
3	-5	2
5	-3	14
15	-1	58

a	b
-1	15	26
-3	5	-2
-5	3	-14
-15	1	-58

(2x -3)(4x + 5)

Factorising Quadratics

where the coefficient of x2 is not 1 ( not a prime)

alternative method

Quadratic Equations

SLIDE NUMBER 29

May 2019

8x2 - 2x - 15

a	b
1	120	119
2	60	58
3	40	37
4	30	26

a	b
5	24	19
6	20	14
8	15	7
10	12	2

8x15 = 120

8x2 + 10x - 12x - 15

2x(4x + 5) - 3(4x + 5)

(2x-3)(4x + 5)

Review of main ideas from above:

Factorise each of the following questions

SLIDE NUMBER 30

May 2019

CONSIDER…

x2 - 12x + 35

x2 - 17x + 30

x2 + 9x + 20

x2 + 19x + 18

x2 + 2x - 48

x2 - 1

3x2 - 15x

4x2 - 64

(x-7)(x-5)

(x-15)(x-2)

(x+4)(x+5)

(x+18)(x+1)

(x+8)(x-6)

(x+1)(x-1)

3x(x-5)

(2x+8)(2x-8)

2x2 + 5x +3

5x2 - 38x +21

6x2 +17x + 12

9x2 +9x - 10

(2x+3)(x+1)

(5x-3)(x-7)

(3x+4)(2x+3)

(3x-2)(3x+5)

Session Objectives

SLIDE NUMBER 31

May 2019

The purpose of the session is to be able to:

Recall the key characteristics of linear and quadratic graphs.

Factorise a quadratic equation including the difference of two squares.

Apply factorising to solving a quadratic equation.

Solve quadratic equations by completing the square.

Solve quadratic equations by using the quadratic formula.

Solve quadratic inequalities.

Solve quadratic equations using an iterative method.

Solving a quadratic equation

Quadratic Equations

SLIDE NUMBER 32

May 2019

Root

(0,-2)

(0, 2)

x2 - 4

(x-2)(x+2)= 0

x = -2 or 2

Solving a Quadratic Equation

Quadratic Equations

SLIDE NUMBER 33

May 2019

Root

(-4,0)

(2, 0)

-(x+4)(x-2)= 0

x = -4 or 2

-x2 - 2x + 8

-(x2 +2x - 8)= 0

Solving Quadratic Equations

Quadratic Equations

SLIDE NUMBER 34

May 2019

x2 + 6x - 16 = 0

(x + 8 )(x - 2 ) = 0

(x + 8 )= 0 and (x - 2 ) = 0

x + 8 = 0

x = -8

and

x - 2 = 0

x = 2

The roots are (2,0) and (-8,0)

Solving Quadratic Equations

- where the coefficient of x2 is not 1

Quadratic Equations

SLIDE NUMBER 35

May 2019

8x2 - 2x – 15 = 0

(2x -3)(4x + 5) = 0

(2x - 3)= 0 and (4x + 5) = 0

2x - 3 = 0

2x = 3

x = 1.5

and

4x + 5 = 0

4x = -5

x = -1.25

The roots are (1.5,0) and (-1.25,0)

Review of main ideas from above:

Solve each of the following questions you previously factorised

SLIDE NUMBER 36

May 2019

CONSIDER…

x2 - 12x + 35=0

x2 - 17x + 30=0

x2 + 9x + 20=0

x2 + 19x + 18=0

x2 + 2x - 48=0

x2 – 1=0

3x2 - 15x=0

4x2 – 64=0

x = 5 or 7

x = 2 or 15

x = -2 or -15

x = -18 or -1

x = -8 or 6

x = 1 or -1

x = 0 or 5

x = 4 or -4

2x2 + 5x +3=0

5x2 - 38x +21=0

6x2 +17x + 12=0

9x2 +9x - 10=0

x = -1.5 or -1

x = 0.6 or 7

x = -1.3 or -1.5

x = -1.6 or 0.6

Session Objectives

SLIDE NUMBER 37

May 2019

The purpose of the session is to be able to:

Recall the key characteristics of linear and quadratic graphs.

Factorise a quadratic equation including the difference of two squares.

Apply factorising to solving a quadratic equation.

Solve quadratic equations by completing the square.

Solve quadratic equations by using the quadratic formula.

Solve quadratic inequalities.

Solve quadratic equations using an iterative method.

Completing the Square

Quadratic Equations

SLIDE NUMBER 38

May 2019

Minimum Point

Minimum point (-3, -21)

Minimum value is -21

X2 + 6x - 12

Turning point

x2 + 6x – 12 = 0

(x + 3)2 – 9 – 12 = 0

(x + 3)2 – 21 = 0

(x + 3)2 = 21

x + 3 = ± 21

x = ± 21 – 3

x = 1.58 or x = -7.58 (2dp)

Solving using Completing the Square

Quadratic Equations

SLIDE NUMBER 39

May 2019

Roots (1.58,0) and (-7.58,0)

Half the coefficient of x

Take away the square of half the coefficient of x

Completed square form

Minimum point (-3, -21)

Minimum value is -21

√

Completing the Square

Quadratic Equations

SLIDE NUMBER 40

May 2019

Turning point

(-1, 9)

Maximum point

-x2 - 2x + 8

Maximum point (-1, 9)

Maximum value is 9

-x2 - 2x + 8 = 0

-(x2 + 2x – 8) = 0

x2 + 2x – 8 = 0

(x + 1)2 – 1 – 8 = 0

(x + 1)2 – 9 = 0

(x + 1)2 = 9

x + 1 = ± 9

x = ± 3 – 1

x = -4 or x = 2

√

Solving using Completing the Square

Quadratic Equations

SLIDE NUMBER 41

May 2019

Roots (-4,0) and (2,0)

Completed square form

Maximum point (-1, 9)

Maximum value is 9

4x2 - 8x + 1 = 0

4(x2 - 2x + 0.25) = 0

x2 - 2x + 0.25 = 0

(x - 1)2 -1 + 0.25 = 0

(x - 1)2 - 0.75 = 0

(x - 1)2 = 0.75

x - 1 = ± 0.75

x = 1 ± 0.75

x = 1.87 (2dp) or x = 0.13 (2dp)

√

Solving using Completing the Square

Quadratic Equations

SLIDE NUMBER 42

May 2019

Roots (1.87,0) and (0.13,0)

Minimum point (1, -0.75)

Minimum value is -0.75

2x2 = 8x +1 1

2x2 - 8x -11 = 0

2(x2 - 4x – 5.5) = 0

x2 - 4x – 5.5 = 0

(x - 2)2 -4 -5.5 = 0

(x - 2)2 -9.5 = 0

(x - 2)2 = 9.5

x - 2 = ± 9.5

x = 2 ± 9.5

x = 5.08 (2dp) or x = -1.08 (2dp)

√

Solving using Completing the Square

Quadratic Equations

SLIDE NUMBER 43

May 2019

Minimum point (2, -9.5)

Minimum value is -9.5

Review of main ideas from above:

Use the completing the square to find the turning point and to solve the following quadratics. (Give answers to 2dp where necessary)

SLIDE NUMBER 44

May 2019

CONSIDER…

1. x2 - 8x = -13	2. 2x2 + 10x + 5 = 0
3. x2 - 3x - 11 = 0	4. 3x2 + 2 = 9x

x = 5.73 (2dp) or x = 2.27 (2dp) Turning point (4,-3)	x = -0.56 (2dp) or x = -4.44 (2dp) Turning point (-2.5,-3.75)
x = 5.14 (2dp) or x = -2.14 (2dp) Turning point (1.5,-13.25)	x = 2.76 (2dp) or x = 0.24 (2dp) Turning point (1.5-1.58)

Session Objectives

SLIDE NUMBER 45

May 2019

The purpose of the session is to be able to:

Recall the key characteristics of linear and quadratic graphs.

Factorise a quadratic equation including the difference of two squares.

Apply factorising to solving a quadratic equation.

Solve quadratic equations by completing the square.

Solve quadratic equations by using the quadratic formula.

Solve quadratic inequalities.

Solve quadratic equations using an iterative method.

The quadratic formula

Quadratic Equations

SLIDE NUMBER 46

May 2019

By completing the square on the quadratic equation

ax2 + bx + c = 0

a quadratic formula can be found which will solve most quadratic equations which do not factorise.

This is an alternative method of solving equations other than by completing the square.

The quadratic formula for solving quadratic equations is

x = -b ± b2 - 4ac

√

The quadratic formula

Quadratic Equations

SLIDE NUMBER 47

May 2019

x = -b ± b2 - 4ac

√

x2 - 2x - 6 = 0

ax2 + bx + c = 0

then a = 1, b = -2 and c = -6

x = -(-2) ± (-2)2 – 4(1)(-6)

2(1)

√

x = 2 ± 4 – -24

√

x = 2 ± 28

√

x = 2 ± 2 7

√

x = 1 ± 7

√

x = 3.645751311 or

x = -1.645751311

x = 3.65 (2dp) or

x = -1.65 (2dp)

The quadratic formula

Quadratic Equations

SLIDE NUMBER 48

May 2019

x = -b ± b2 - 4ac

√

3 x2 + 4x - 2 = 0

ax2 + bx + c = 0

then a = 3, b = 4 and c = -2

x = -(4) ± (4)2 – 4(3)(-2)

2(3)

√

x = -4 ± 16 – -24

√

x = -4± 40

√

x = -4 ±2 10

√

x = 0.3874258867 or

x = -1.72075922

x = -2 ± 10

√

x = 0.39 (2dp) or

x = -1.72 (2dp)

Review of main ideas from above:

Use the quadratic formula to solve the following quadratics

SLIDE NUMBER 49

May 2019

CONSIDER…

1. 3x2 + 6x - 7 = 0	2. x2 + 3x + 1 = 0
3. 8x2 - 6x + 1 = 0	4. 4x2 + 7x - 6 = 0

x = 0.83 (2dp) or x = -2.83 (2dp)	2. x = -0.38 (2dp) or x = -2.62 (2dp)
3. x = 0.5 or x = 0.25	x = 0.63 (2dp) or x = -2.38 (2dp)

Session Objectives

SLIDE NUMBER 50

May 2019

The purpose of the session is to be able to:

Recall the key characteristics of linear and quadratic graphs.

Factorise a quadratic equation including the difference of two squares.

Apply factorising to solving a quadratic equation.

Solve quadratic equations by completing the square.

Solve quadratic equations by using the quadratic formula.

Solve quadratic inequalities.

Solve quadratic equations using an iterative method.

Solving Quadratic inequalities

Quadratic Equations

SLIDE NUMBER 51

May 2019

x2 + 2x -15

(x+5)(x-3)

Roots: (-5,0)and (3,0)

If …

x2 + 2x -15 ≥ 0

above x-axis

x ≥ 3 and x ≤ -5

Solving Quadratic inequalities

Quadratic Equations

SLIDE NUMBER 52

May 2019

x2 + 2x -15

(x+5)(x-3)

Roots: (-5,0)and (3,0)

If …

x2 + 2x -15 ≤ 0

below x-axis

-5 ≤ x ≤ 3

Solving Quadratic inequalities

Quadratic Equations

SLIDE NUMBER 53

May 2019

x2 + 12x + 82 > 22 – 4x

x2 + 16x + 60 > 0

(x + 6)(x + 10)

Roots: (-6,0)and (-10,0)

x2 + 16x + 60 > 0

above x-axis

x > -6 and x < -10

Solving Quadratic inequalities

Quadratic Equations

SLIDE NUMBER 54

May 2019

5x2 < 80

x2 < 16

x2 - 16 < 0

(x + 4)(x - 4)

Roots: (-4,0)and (4,0)

x2 - 16 < 0

below x-axis

-4 < x < 4

{x: -4 < x < 4}

Review of main ideas from above:

Use the quadratic formula to solve the following quadratics

SLIDE NUMBER 55

May 2019

CONSIDER…

1. x2 - 2x - 48 ≥ 0	2. x2 - 49 ≤ 0
3. 24 < 10x - x2	4. x2 + 7x - 10 > 4x

x ≤ -6 and x ≥ 8	2. -7 ≤ x ≤ 7
3. 4 < x < 6	x > 2 and x < -5

Session Objectives

SLIDE NUMBER 56

May 2019

The purpose of the session is to be able to:

Recall the key characteristics of linear and quadratic graphs.

Factorise a quadratic equation including the difference of two squares.

Apply factorising to solving a quadratic equation.

Solve quadratic equations by completing the square.

Solve quadratic equations by using the quadratic formula.

Solve quadratic inequalities.

Solve quadratic equations using an iterative method.

Solving using an iterative method

Quadratic Equations

SLIDE NUMBER 57

May 2019

Use the iteration formula xn+1 = 2 + 1

to find solutions to x2 - 2x = 1 to 3dp. Use the starting value x0 = 3

Press 3 then =

2 + 1

ANS

x1 = 2.3333

x2 = 2.4286

x3 = 2.4118 x4 = 2.4146

x5 = 2.4141

x6 = 2.4142

x25 = 2.4142

x = 2.414 to 3dp

Review of main ideas from above:

SLIDE NUMBER 58

May 2019

CONSIDER…

Use the iteration formula xn+1 = 1 - 2

to find solutions to x2 + 2x = 1 to 3dp. Use the starting value x0 = 1

Use the iteration formula xn+1 = 3xn + 1

to find solutions to -2x2 + 3x + 1 = 0 to 1dp. Use the starting value x0 = 1

Use the iteration formula xn+1 = 2x2n + 11

to find solutions to 2x2 = 11 to 3dp. Use the starting value x0 = 3

√

4xn

x = -2.414

x = 1.8

x = 2.345

Session Objectives

SLIDE NUMBER 59

May 2019

The purpose of the session is to be able to:

Recall the key characteristics of linear and quadratic graphs.

Factorise a quadratic equation including the difference of two squares.

Apply factorising to solving a quadratic equation.

Solve quadratic equations by completing the square.

Solve quadratic equations by using the quadratic formula.

Solve quadratic inequalities.

Solve quadratic equations using an iterative method.

SLIDE NUMBER 60

May 2019

CONSIDER…

End of Presentation

At this point it would be advisable to go back over the presentation. Ensure that you are fully able to deal accurately and effectively with each session objective.

You should supplement the content of this session with suitable reading, research and discussion with others.

End of presentation

Rebecca Wigfull

SLIDE NUMBER 61

May 2019

GCSE Mathematics – Quadratic Equations