Calculus Homework
Algebra Practice Problems for Precalculus and Calculus
Solve the following equations for the unknown x :
1. 5 = 7x − 16
2. 2x − 3 = 5 − x
3. 12 (x − 3) + x = 17 + 3(4 − x )
4. 5x = 2
x −3
Multiply the indicated polynomials and simplify.
5. (4x − 1)(−3x + 2)
6. (x − 1)(x 2 + x + 1)
7. (x + 1)(x 2 − x + 1)
8. (x − 2)(x + 2)
9. (x − 2)(x − 2)
10. (x 3 + 2x − 1)(x 3 − 5x 2 + 4)
Find the domain of each of the following functions in 11-15.
11. f (x ) = √
1 + x
12. f (x ) = 11+x
13. f (x ) = 1√ x
14. f (x ) = 1√ 1+x
15. f (x ) = 1 1+x 2
16. Given that f (x ) = x 2 − 3x + 4, find and simplify f (3), f (a), f (−t ), and f (x 2 + 1).
Factor the following quadratics
17. x 2 − x − 20
18. x 2 − 10x + 21
19. x 2 + 10x + 16
20. x 2 + 8x − 105
21. 4x 2 + 11x − 3
22. −2x 2 + 7x + 15
23. x 2 − 2
1
Solve the following quadratic equations in three ways: 1) factor, 2) quadratic formula, 3) complete the square
24. x 2 + 6x − 16 = 0
25. −x 2 − 3x − 2 = 0
26. 2x 2 + 2x − 4 = 0
Solve the following smorgasbord of equations and inequalities
27. √
x = √
2x − 1
28. √
x 2 − 3 = √
2x
29. |x − 5| = 4
30. 2x + 4 ≥ 3
31. −2x + 4 ≥ 3
32. x +4x −3 = 2
33. x 2 − x − 2 > 0
Add/Subtract the following rational expressions:
34. xx +2 + 3
x −4
35. x 2+1
(x −1)(x −2) − x 3
x −3
Simplify the following rational expressions (if possible):
36. x 2+x −2 x 2−1
37. x 2+5x +6
x 2−3x +2
38.
x x +2 + 3
x +1 x −1
Solutions
1. Given that 5 = 7x − 16, add 16 to both sides to get 7x = 21. Now divide both sides by 7 to get x = 3. Checking, we see that 7(3) − 16 = 21 − 16 = 5.
2. Given that 2x − 3 = 5 − x , add x to both sides and then add 3 to both sides to get 3x = 8. Now divide both sides by 3 to get x = 8/3 = 2.6̄. Checking, we see that 2(8/3) − 3 = 163 − 3 =
16 3 −
9 3 =
7 3 and 5 − (8/3) =
15 3 −
8 3 =
7 3 .
3. Given that 12 (x − 3) + x = 17 + 3(4 − x ), we first simplify the left and right hand sides using the distributive property to get 1 2 x −
3 2 + x = 17 + 12 − 3x . Combining like terms on both sides gives
3 2 x −
3 2 = 29 − 3x . Now we add 3x and
3 2 to both
sides, obtaining 92 x = 61 2 . Dividing both sides by
9 2 (or multiplying both sides by
2 9 ) gives x =
61 2 ·
2 9 =
61 9 . Checking we see
that L H S = 12 ( 61 9 −
27 9 ) +
61 9 =
1 2
34 9 +
61 9 =
17 9 +
61 9 =
78 9 and R H S = 17 + 3(
36 9 −
61 9 ) = 17 + 3 ·
(
−25 9
)
= 17 − 759 = 153
9 − 75 9 =
78 9 .
2