Factoring FOIL, Graphing Parabolas, and Solving Quadratics – Answer Key| 1

Math 101: Factoring FOIL, Graphing Parabolas, and Solving Quadratics

Practice Problem Set – Answer Key

1. What is the shape of the graph of an equation whose highest degree is 2?

Answer: parabola

2. Which equation could have a parabolic graph?

a) 𝑦 = 𝑥3 + 1

b) 𝑦 = 3𝑥2 − 2

c) 𝑦 = 3𝑥 − 6

d) 𝑦 = 2𝑥5 − 𝑥2 + 2

Answer: b) it is the only equation with an even highest degree

3. Is the graph of 𝑦 = 𝑥3 + 2𝑥2– 1 a parabola?

Answer: No, the highest degree (3) is odd so it cannot be parabolic

4. What is the vertex of the parabola 𝑦 = 𝑥2 + 4𝑥 + 2?

Answer: axis of symmetry: 𝑥 =−𝑏

2𝑎

𝑥 =−4

2(1)

𝑥 =−4

2= −2

vertex: 𝑦 = 𝑥2 + 4𝑥 + 2

𝑦 = (−2)2 + 4(−2) + 2

𝑦 = 4 − 8 + 2 = −2

(−2, −2)

5. What is the vertex of the parabola 𝑦 = 3𝑥2 + 18𝑥 − 3

Answer: axis of symmetry: 𝑥 =−𝑏

2𝑎

𝑥 =−18

2(3)

𝑥 =−18

6= −3

vertex: 𝑦 = 3𝑥2 + 18𝑥 − 3

𝑦 = 3(−3)2 + 18(−3) − 3

𝑦 = 27 + (−54) − 3 = −30

(−3, −30)

Factoring FOIL, Graphing Parabolas, and Solving Quadratics – Answer Key| 2

6. What is the vertex of the parabola 𝑦 = −𝑥2 − 2𝑥 + 1?

Answer: axis of symmetry: 𝑥 =−𝑏

2𝑎

𝑥 =−(−2)

2(−1)

𝑥 =2

−2= −1

vertex: 𝑦 = − 𝑥2– 2𝑥 + 1

𝑦 = − (−1)2 − 2(−1) + 1

𝑦 = −1 − (−2) + 1 = 2

(−1,2)

7. Determine if the vertex of the parabola is a maximum or minimum. 𝑦 = 2𝑥2 + 𝑥 − 1

Answer: The vertex is a minimum. Since the “a” value is positive, the parabola opens upward

making the vertex the lowest point.

8. Is the vertex of 𝑦 = − 𝑥2 + 2𝑥 + 3 a minimum or maximum?

Answer: The vertex is a maximum. Since the “a” value is negative, the parabola opens

downward making the vertex the highest point.

9. Which real-life relationship would be graphed as a parabola, the growth of a child, the distance

a car drives, or the height of a tossed ball?

Answer: The height of a tossed ball would be graphed with a parabola because the path of

launched objects have a symmetric rise and fall, in terms of height.

Factoring FOIL, Graphing Parabolas, and Solving Quadratics – Answer Key| 3

10. Which is the graph of 𝑦 = −2(𝑥 + 2)2 − 1?

a) c)

b) d)

Answer: c)

The graph of 𝑦 = −2(𝑥 + 2)2– 1 will be a parabola that opens down, has been shifted left 2

units, and down 1 unit.

Factoring FOIL, Graphing Parabolas, and Solving Quadratics – Answer Key| 4

11. Which is the graph of 𝑦 = 3(𝑥 − 1)2 + 4?

a) c)

b) d)

Answer: b)

This graph has been shrunk and shifted right 1 unit and up 4 units from a standard, 𝑦 = 𝑥2

parabola.

Factoring FOIL, Graphing Parabolas, and Solving Quadratics – Answer Key| 5

12. When is the FOIL method used and for what does it stand?

Answer: The FOIL method is used to multiply binomials. The letters stand for first, outer, inner,

and last.

13. Multiply (𝑥 − 2)(𝑥 + 3).

Answer:

(𝑥 − 2)(𝑥 + 3) (𝑥 ⋅ 𝑥) + (𝑥 ⋅ 3) + (−2 ⋅ 𝑥) + (−2 ⋅ 3)

𝑥2 + 3𝑥 − 2𝑥 + (−6)

𝑥2 + 𝑥 − 6

14. Multiply (𝑥 + 4)(𝑥 − 1)

Answer:

(𝑥 + 4)(𝑥 − 1) (𝑥 ⋅ 𝑥) + (𝑥 ⋅ −1) + (4 ⋅ 𝑥) + (4 ⋅ −1)

𝑥2 − 𝑥 + 4𝑥 − 4

𝑥2 + 3𝑥 − 4

15. Multiply (𝑥 − 5)(𝑥 − 3)

Answer:

(𝑥 − 5)(𝑥 − 3) (𝑥 ⋅ 𝑥) + (𝑥 ⋅ −3) + (−5 ⋅ 𝑥) + (−5 ⋅ −3)

𝑥2 − 3𝑥 − 5𝑥 + 15

𝑥2 − 8𝑥 + 15

16. Factor 𝑥2 + 𝑥 – 2, using the ac- method.

Answer:

(𝑥 − 1)(𝑥 + 2)

Since the “a” term is 1, we can simply ask what two values will have a sum of “b”, 1, and a

product of “c”, -2. The only factors of “c” that add to “b” are -1 and 2. So our factors are (x – 1)

and (x + 2)

Factoring FOIL, Graphing Parabolas, and Solving Quadratics – Answer Key| 6

17. Factor 𝑥2 + 7𝑥 + 12 using the area method.

Answer:

(𝑥 + 3)(𝑥 + 4)

What two numbers have a sum of “b” and product of “ac”?

𝑏 = 7

𝑎𝑐 = 1 ⋅ 12 = 12

3 + 4 = 7

3 ⋅ 4 = 12

𝑏1 = 3

𝑏2 = 4

GCF GCF 𝑥 4

𝑎𝑥2 𝑏1𝑥 𝑥 𝑥2 4𝑥

𝑏2𝑥 𝑐 3 3𝑥 12

(𝑥 + 4)(𝑥 + 3)

18. Factor 𝑥2– 9 using the area method.

Answer:

(𝑥 + 3)(𝑥 − 3)

The equation has a “b” value of zero, so we fill that in to factor: 𝑥2 + 0𝑥 − 9

What two values have a sum of zero and product of -9?

𝑏1 = 3

𝑏2 = −3

GCF GCF 𝑥 3

𝑎𝑥2 𝑏1𝑥 𝑥 𝑥2 3𝑥

𝑏2𝑥 𝑐 −3 −3𝑥 −9

(𝑥 − 3)(𝑥 + 3)

Factoring FOIL, Graphing Parabolas, and Solving Quadratics – Answer Key| 7

19. Factor 𝑥2 − 36

Answer:

(𝑥 + 6)(𝑥 − 6)

If both the “a” and “c” terms are perfect squares with “c” being subtracted from “a”, and there

is no “b” term, the factors will always be in the form of (𝑥 + √𝑐)(𝑥 − √𝑐), as this gives b-

values that cancel each other.

20. Factor 𝑥2– 81.

Answer:

(𝑥 + 9)(𝑥 − 9)

The “a” and “c” terms are both perfect squares, with no “b” term and “c” being subtracted from

“a”. The factors are in the form (𝑥 + √𝑐) and (𝑥 − √𝑐).

21. Factor 𝑥2 − 8𝑥 + 16.

Answer:

(𝑥 − 4)(𝑥 − 4)

What two numbers have a sum of “b”, -8, and a product of “ac”, 16?

−4 + −4 = −8 𝑎𝑛𝑑 − 4 ⋅ −4 = 16

Since -4 and -4 are the only factors of 16 that add up to -8, our factors are (x – 4)(x – 4).

Factoring FOIL, Graphing Parabolas, and Solving Quadratics – Answer Key| 8

22. Factor 2𝑥2 − 5𝑥 − 12

Answer:

(2𝑥 + 3)(𝑥 − 4)

What two numbers add to -5 and have a product of -24, “ac”?

−8 + 3 = −5 and − 8 ⋅ 3 = −24

𝑏1 = −8 𝑏2 = 3

GCF GCF 2𝑥 3

𝑎𝑥2 𝑏1𝑥 𝑥 2𝑥2 3𝑥

𝑏2𝑥 𝑐 −4 −8𝑥 −12

Factors: (2𝑥 + 3)(𝑥 − 4)

23. Factor 6𝑥2 + 7𝑥 − 3.

Answer:

(3𝑥 − 1)(2𝑥 + 3)

What factors of -18, “ac”, have a sum of 7, “b”? Nine and -2 add for a sum of 7 and have a

product of -18, so they are our b-values.

GCF GCF 3𝑥 −1

𝑎𝑥2 𝑏1𝑥 2𝑥 6𝑥2 −2𝑥

𝑏2𝑥 𝑐 3 9𝑥 −3

Factors: (3𝑥 − 1)(2𝑥 + 3)

Factoring FOIL, Graphing Parabolas, and Solving Quadratics – Answer Key| 9

24. Factor 3𝑥2 + 𝑥 − 2.

Answer:

(3𝑥 − 2)(𝑥 + 1)

What factors of -6, “ac”, have a sum of 1, “b”? Three and -2 add for a sum of 1 and have a

product of -6, so they are our b-values.

GCF GCF 3𝑥 −2

𝑎𝑥2 𝑏1𝑥 𝑥 3𝑥2 −2𝑥

𝑏2𝑥 𝑐 1 3𝑥 −2

Factors: (3𝑥 − 2)(𝑥 + 1)

25. Solve by completing the square: 𝑥2 + 4𝑥 − 5 = 0

Answer:

(𝑏

2)

2

= (4

2)

2

= 4

𝑥2 + 4𝑥 − 5 = 0

+5 + 5

𝑥2 + 4𝑥 + (𝑏

2)

2

= 5 + (𝑏

2)

2

𝑥2 + 4𝑥 + 4 = 5 + 4 (𝑥 + 2)2 = 9

√(𝑥 + 2)2 = √9

(𝑥 + 2) = ±3

−2 − 2

𝑥 = 3 − 2

𝑥 = 1

𝑥 = −3 − 2

𝑥 = −5

Factoring FOIL, Graphing Parabolas, and Solving Quadratics – Answer Key| 10

26. Solve by completing the square: 𝑥2 + 8𝑥 + 12 = 0

Answer:

(𝑏

2)

2

= (8

2)

2

= 16

𝑥2 + 8𝑥 + 12 = 0

−12 − 12

𝑥2 + 8𝑥 = −12

𝑥2 + 8𝑥 + (𝑏

2)

2

= −12 + (𝑏

2)

2

𝑥2 + 8𝑥 + 16 = −12 + 16 (𝑥 + 4)2 = 4

√(𝑥 + 4)2 = √4

𝑥 + 4 = ±2

−4 − 4

𝑥 = 2 − 4

𝑥 = −2

𝑥 = −2 − 4

𝑥 = −6

27. Solve by completing the square: 𝑥2 − 6 = 6𝑥 + 3

Answer:

(𝑏

2)

2

= (−6

2)

2

= 9

𝑥2 − 6 = 6𝑥 + 3

−6𝑥 − 6𝑥

𝑥2 − 6𝑥 − 6 = 3

+6 + 6

𝑥2 − 6𝑥 = 9

𝑥2 − 6𝑥 + 9 = 9 + 9

(𝑥 − 3)2 = 18

√(x − 3)2 = √18

𝑥 − 3 = ±3√2

+3 + 3

𝑥 = 3 + 3√2

𝑥 = 3 − 3√2

Factoring FOIL, Graphing Parabolas, and Solving Quadratics – Answer Key| 11

28. Find the solutions of 𝑥2 + 2 = 10𝑥– 7 by completing the square.

Answer:

(𝑏

2)

2

= (−10

2)

2

= 25

𝑥2 + 2 = 10𝑥 − 7

−2 − 2

𝑥2 = 10𝑥 − 9

−10𝑥 − 10𝑥

𝑥2 − 10𝑥 = −9

𝑥2 − 10𝑥 + (𝑏

2)

2

= −9 + (𝑏

2)

2

𝑥2 − 10𝑥 + 25 = −9 + 25

(𝑥 − 5)2 = 16

√(𝑥 − 5)2 = 16

𝑥 − 5 = ±4

+5 + 5

𝑥 = 5 + 4

𝑥 = 9

𝑥 = 5 − 4

x = 1

29. Find the solutions of 𝑥2 − 3𝑥 + 2 = 0

Answer:

𝑥2 − 3𝑥 + 2 = 0 (𝑥 − 2)(𝑥 − 1) = 0

𝑥 − 2 = 0

+2 + 2

𝑥 = 2

𝑥 − 1 = 0

+1 + 1

𝑥 = 1

30. Find the solutions of 𝑥2 + 2𝑥 − 8 = 0

Answer:

𝑥2 + 2𝑥 − 8 = 0 (𝑥 + 4)(𝑥 − 2) = 0

𝑥 + 4 = 0

−4 − 4

𝑥 = −4

𝑥 − 2 = 0

+2 + 2

𝑥 = 2

Factoring FOIL, Graphing Parabolas, and Solving Quadratics – Answer Key| 12

31. Find the roots of 𝑓(𝑥) = 𝑥2 + 6𝑥 + 8.

Answer:

𝑓(𝑥) = 𝑥2 + 6𝑥 + 8

0 = 𝑥2 + 6𝑥 + 8

0 = (𝑥 + 4)(𝑥 + 2)

0 = 𝑥 + 4

−4 − 4

−4 = 𝑥

0 = 𝑥 + 2

−2 − 2

−2 = 𝑥

32. Find the roots of 𝑓(𝑥) = 𝑥2 − 8𝑥 + 15

Answer:

𝑓(𝑥) = 𝑥2 − 8𝑥 + 15

0 = 𝑥2 − 8𝑥 + 15

0 = (𝑥 − 3)(𝑥 − 5)

0 = 𝑥 − 3

+3 + 3

3 = 𝑥

0 = 𝑥 − 5

+5 + 5

5 = 𝑥

33. Use the quadratic formula to find the solutions of 𝑦 = −4𝑥2 + 7𝑥 − 2.

Answer:

𝑥 =−𝑏 ± √𝑏2 − 4𝑎𝑐

2𝑎

𝑎 = −4, 𝑏 = 7, 𝑐 = −2

𝑥 =−7 ± √72 − 4(−4)(−2)

2(−4)

𝑥 =−7 ± √49 − 32

−8

𝑥 =−7 ± √17

−8

𝑥 =−7 + √17

−8

𝑥 =−7 − √17

−8=

−1(7 + √17)

−8=

7 + √17

8

Factoring FOIL, Graphing Parabolas, and Solving Quadratics – Answer Key| 13

34. Use the quadratic formula to find the solutions of 𝑦 = 3𝑥2 − 8𝑥 + 3

Answer:

𝑥 =−𝑏 ± √𝑏2 − 4𝑎𝑐

2𝑎

𝑎 = 3, 𝑏 = −8, 𝑐 = 3

𝑥 =−(−8) ± √(−8)2 − 4(3)(3)

2(3)

𝑥 =8 ± √64 − 36

6

𝑥 =8 ± √28

6

𝑥 =8 + √28

6

𝑥 =8 − √28

6

35. Use the quadratic formula to find the solutions of 𝑦 = 𝑥2 + 6𝑥 + 3

Answer:

𝑥 =−𝑏 ± √𝑏2 − 4𝑎𝑐

2𝑎

𝑎 = 1, 𝑏 = 6, 𝑐 = 3

𝑥 =−6 ± √62 − 4(1)(3)

2(1)

𝑥 =−6 ± √24

2=

−6 ± 2√6

2

𝑥 =−6 + 2√6

2

𝑥 = −3 + √6

𝑥 =−6 − 2√6

2

𝑥 = −3 − √6

36. Find the discriminant of −2𝑥2 − 3𝑥 + 5 = 0.

Answer:

𝐷 = √𝑏2 − 4𝑎𝑐

𝐷 = √(−3)2 − 4(−2)(5)

𝐷 = √9 + 40

𝐷 = √49 = 7

Factoring FOIL, Graphing Parabolas, and Solving Quadratics – Answer Key| 14

37. Find the discriminant of 3𝑥2 − 𝑥 − 6 = 0.

Answer:

𝐷 = √𝑏2 − 4𝑎𝑐

𝐷 = √(−1)2 − 4(3)(−6)

𝐷 = √1 + 72

𝐷 = √73

38. Find the discriminant of 𝑥2 − 10𝑥 + 3 = 0.

Answer:

𝐷 = √𝑏2 − 4𝑎𝑐

𝐷 = √(−10)2 − 4(1)(3)

𝐷 = √100 − 12

𝐷 = √88 = 2√22

39. Solve (𝑥 − 2)2 + 5𝑥 = 5

Answer:

(𝑥 − 2)2 + 5𝑥 = 5 (𝑥 − 2)(𝑥 − 2) + 5𝑥 = 5

𝑥2 − 4𝑥 + 4 + 5𝑥 = 5

𝑥2 + 𝑥 + 4 = 5

−5 − 5

𝑥2 + 𝑥 − 1 = 0

𝑥 =−𝑏 ± √𝑏2 − 4𝑎𝑐

2𝑎

𝑥 =−1 ± √12 − 4(1)(−1)

2(1)

𝑥 =−1 ± √1 + 4

2

𝑥 =−1 ± √5

2

𝑥 =−1 + √5

2

𝑥 =−1 − √5

2

Factoring FOIL, Graphing Parabolas, and Solving Quadratics – Answer Key| 15

40. Solve (𝑥 + 1)2 − 3𝑥 = 7.

Answer:

(𝑥 + 1)2 − 3𝑥 = 7 (𝑥 + 1)(𝑥 + 1) − 3𝑥 = 7

𝑥2 + 2𝑥 + 1 − 3𝑥 = 7

𝑥2 − 𝑥 + 1 = 7

𝑥2 − 𝑥 − 6 = 0 (𝑥 − 3)(𝑥 + 2) = 0

𝑥 − 3 = 0

𝑥 = 3

𝑥 + 2 = 0

𝑥 = −2

41. Solve (𝑥 − 3)2 = 10 − 4𝑥.

Answer:

(𝑥 − 3)(𝑥 − 3) = 10 − 4𝑥

𝑥2 − 6𝑥 + 9 = 10 − 4𝑥

𝑥2 − 2𝑥 − 1 = 0

𝑥2 − 2𝑥 = 1

𝑥2 − 2𝑥 + (𝑏

2)

2

= 1 + (𝑏

2)

2

𝑥2 − 2𝑥 + 1 = 1 + 1

(𝑥 − 1)2 = 2

√(𝑥 − 1)2 = √2

𝑥 − 1 = ±√2

+1 + 1

𝑥 = 1 + √2

𝑥 = 1 − √2