9-5 Solving Quadratic Equations by Using the Quadratic Formula
Solve each equation by using the Quadratic Formula. Round to the nearest tenth if necessary.
2
1. x − 2x − 15 = 0
SOLUTION: For this equation, a = 1, b = –2, and c = –15.
The solutions are 5 and –3.
2
2. x − 10x + 16 = 0
SOLUTION: For this equation, a = 1, b = –10, and c = 16.
The solutions are 8 and 2.
2
3. x − 8x = −10
SOLUTION: Write the equation in standard form.
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Page 1
9-5 Solving Quadratic Equations by Using the Quadratic Formula
The solutions are 8 and 2.
2
3. x − 8x = −10
SOLUTION: Write the equation in standard form.
For this equation, a = 1, b = –8, and c = 10.
The solutions are 6.4 and 1.6.
2
4. x + 3x = 12
SOLUTION: Write the equation in standard form.
For this equation, a = 1, b = 3, and c = –12.
The solutions are 2.3 and –5.3.
2
5. 10x Manual
− 31x- +Powered
15 = 0by Cognero
eSolutions
SOLUTION: For this equation, a = 10, b = –31, and c = 15.
Page 2
9-5 Solving Quadratic Equations by Using the Quadratic Formula
The solutions are 2.3 and –5.3.
2
5. 10x − 31x + 15 = 0
SOLUTION: For this equation, a = 10, b = –31, and c = 15.
The solutions are 2.5 and 0.6.
2
6. 5x + 5 = −13x
SOLUTION: Write the equation in standard form.
For this equation, a = 5, b = 13, and c = 5.
The solutions are –0.5 and –2.1.
Solve each equation. State which method you used.
2
7. 2x + 11x − 6 = 0
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Page 3
SOLUTION: Solve by factoring:
In this trinomial, a = 2, b = 11, and c = –6, so m + p is positive and mp is negative. Therefore, m and p must have
9-5 Solving Quadratic Equations by Using the Quadratic Formula
The solutions are –0.5 and –2.1.
Solve each equation. State which method you used.
2
7. 2x + 11x − 6 = 0
SOLUTION: Solve by factoring:
In this trinomial, a = 2, b = 11, and c = –6, so m + p is positive and mp is negative. Therefore, m and p must have
different signs. List the factors of 2(–6) or –12 and identify the factors with a sum of 11.
Factors of –12
Sum 1, –12
–11
–1, 12
11
2, –6
–4
–2, 6
4
3, –4
–1
–3, 4
1
The correct factors are –1 and 12. So, use m = -1 and p = 12.
Solve the equation using the Zero Product Property
The solutions are –6 and
.
2
8. 2x − 3x − 6 = 0
SOLUTION: Solve using the quadratic formula.
For this equation, a = 2, b = –3, and c = –6.
Quadratic Formula:
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The solutions are 2.6 and –1.1.
Page 4
The solutions
are –6Equations
and . by Using the Quadratic Formula
9-5 Solving
Quadratic
2
8. 2x − 3x − 6 = 0
SOLUTION: Solve using the quadratic formula.
For this equation, a = 2, b = –3, and c = –6.
Quadratic Formula:
The solutions are 2.6 and –1.1.
2
9. 9x = 25
SOLUTION: Solve by factoring.
Write the equation in standard form.
Factoring:
The solutions are
.
2
10. x − 9x = −19
SOLUTION: Solve by using the quadratic formula.
Write the equation in standard form.
For this equation, a = 1, b = –9, and c = 19.
Quadratic Formula:
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Page 5
9-5 Solving
Quadratic
by Using the Quadratic Formula
The solutions
are Equations
.
2
10. x − 9x = −19
SOLUTION: Solve by using the quadratic formula.
Write the equation in standard form.
For this equation, a = 1, b = –9, and c = 19.
Quadratic Formula:
The solutions are 5.6 and 3.4.
State the value of the discriminant for each equation. Then determine the number of real solutions of the
equation.
2
11. x − 9x + 21 = 0
SOLUTION: For this equation, a = 1, b = –9, and c = 21.
The discriminant is –3.
Since the discriminant is negative, the equation has no real solutions.
2
12. 2x − 11x + 10 = 0
SOLUTION: For this equation, a = 2, b = –11, and c = 10.
The discriminant is 41.
Since the discriminant is positive, the equation has two real solutions.
2
13. 9x + 24x = −16
SOLUTION: eSolutions
Manual
- Powered in
by standard
Cognero
Write
the equation
form.
Page 6
9-5 Solving
Quadratic
Equations by Using the Quadratic Formula
The discriminant
is 41.
Since the discriminant is positive, the equation has two real solutions.
2
13. 9x + 24x = −16
SOLUTION: Write the equation in standard form.
For this equation, a = 9, b = 24, and c = 16.
The discriminant is 0.
Since the discriminant is 0, the equation has one real solution.
2
14. 3x − x = 8
SOLUTION: Write the equation in standard form.
For this equation, a = 3, b = –1, and c = –8.
The discriminant is 97.
Since the discriminant is positive, the equation has two real solutions.
15. TRAMPOLINE Eva springs from a trampoline to dunk a basketball. Her height h in feet can be modeled by the
2
equation h = –16t + 22.3t + 2, where t is time in seconds. Use the discriminant to determine if Eva will reach a
height of 10 feet. Explain.
SOLUTION: Write the equation in standard form.
For this equation, a = –16, b = 22.3, and c = –8.
The discriminant is −14.91.
Since the discriminant is negative, the equation has no real solutions. Thus, Eva will not reach a height of 10 feet.
Solve each equation by using the Quadratic Formula. Round to the nearest tenth if necessary.
2
16. 4x + 5x − 6 = 0
SOLUTION: For this equation, a = 4, b = 5, and c = –6.
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Page 7
9-5 Solving
Quadratic
Equations by Using the Quadratic Formula
The discriminant
is −14.91.
Since the discriminant is negative, the equation has no real solutions. Thus, Eva will not reach a height of 10 feet.
Solve each equation by using the Quadratic Formula. Round to the nearest tenth if necessary.
2
16. 4x + 5x − 6 = 0
SOLUTION: For this equation, a = 4, b = 5, and c = –6.
The solutions are
and –2.
2
17. x + 16 = 0
SOLUTION: For this equation, a = 1, b = 0, and c = 16.
The discriminant is negative, so the equation has no real solutions, ø.
2
18. 6x − 12x + 1 = 0
SOLUTION: For this equation, a = 6, b = –12, and c = 1.
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Page 8
9-5 Solving Quadratic Equations by Using the Quadratic Formula
The discriminant is negative, so the equation has no real solutions, ø.
2
18. 6x − 12x + 1 = 0
SOLUTION: For this equation, a = 6, b = –12, and c = 1.
The solutions are 1.9 and 0.1.
2
19. 5x − 8x = 6
SOLUTION: Write the equation in standard form.
For this equation, a = 5, b = –8, and c = –6.
The solutions are 2.2 and –0.6.
2
20. 2x − 5x = −7
SOLUTION: Write the equation in standard form.
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For this equation, a = 2, b = –5, and c = 7.
Page 9
9-5 Solving Quadratic Equations by Using the Quadratic Formula
The solutions are 2.2 and –0.6.
2
20. 2x − 5x = −7
SOLUTION: Write the equation in standard form.
For this equation, a = 2, b = –5, and c = 7.
The discriminant is negative, so the equation has no real solutions, Ø.
2
21. 5x + 21x = −18
SOLUTION: Write the equation in standard form.
For this equation, a = 5, b = 21, and c = 18.
The solutions are
and –3.
2
22. 81x = 9
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SOLUTION: Write the equation in standard form.
Page 10
9-5 Solving
Quadratic
The solutions
are Equations
and –3. by Using the Quadratic Formula
2
22. 81x = 9
SOLUTION: Write the equation in standard form.
For this equation, a = 81, b = 0, and c = –9.
The solutions are
.
2
23. 8x + 12x = 8
SOLUTION: Write the equation in standard form.
For this equation, a = 8, b = 12, and c = –8.
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Page 11
9-5 Solving
Quadratic
by Using the Quadratic Formula
The solutions
are Equations
.
2
23. 8x + 12x = 8
SOLUTION: Write the equation in standard form.
For this equation, a = 8, b = 12, and c = –8.
The solutions are 0.5 and –2.
2
24. 4x = −16x − 16
SOLUTION: Write the equation in standard form.
For this equation, a = 4, b = 16, and c = 16.
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The solution is –2.
Page 12
9-5 Solving Quadratic Equations by Using the Quadratic Formula
The solutions are 0.5 and –2.
2
24. 4x = −16x − 16
SOLUTION: Write the equation in standard form.
For this equation, a = 4, b = 16, and c = 16.
The solution is –2.
2
25. 10x = −7x + 6
SOLUTION: Write the equation in standard form.
For this equation, a = 10, b = 7, and c = –6.
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The solutions are 0.5 and –1.2.
Page 13
9-5 Solving Quadratic Equations by Using the Quadratic Formula
The solution is –2.
2
25. 10x = −7x + 6
SOLUTION: Write the equation in standard form.
For this equation, a = 10, b = 7, and c = –6.
The solutions are 0.5 and –1.2.
2
26. −3x = 8x − 12
SOLUTION: Write the equation in standard form.
For this equation, a = –3, b = –8, and c = 12.
The solutions are –3.7 and 1.1.
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2
27. 2x = 12x − 18
Page 14
9-5 Solving Quadratic Equations by Using the Quadratic Formula
The solutions are 0.5 and –1.2.
2
26. −3x = 8x − 12
SOLUTION: Write the equation in standard form.
For this equation, a = –3, b = –8, and c = 12.
The solutions are –3.7 and 1.1.
2
27. 2x = 12x − 18
SOLUTION: Write the equation in standard form.
For this equation, a = 2, b = –12, and c = 18.
The solution is 3.
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Page 15
28. AMUSEMENT PARKS The Demon Drop at Cedar Point in Ohio takes riders to the top of a tower and drops
2
them 60 feet. A function that approximates this ride is h = −16t + 64t − 60, where h is the height in feet and t is the
9-5 Solving Quadratic Equations by Using the Quadratic Formula
The solutions are –3.7 and 1.1.
2
27. 2x = 12x − 18
SOLUTION: Write the equation in standard form.
For this equation, a = 2, b = –12, and c = 18.
The solution is 3.
28. AMUSEMENT PARKS The Demon Drop at Cedar Point in Ohio takes riders to the top of a tower and drops
2
them 60 feet. A function that approximates this ride is h = −16t + 64t − 60, where h is the height in feet and t is the
time in seconds. About how many seconds does it take for riders to drop 60 feet?
SOLUTION: 2
−16t + 64t − 60 = 0
For this equation, a = –16, b = 64, and c = –60.
It takes 2.5 seconds for the riders to drop from 60 feet to 0 feet.
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Solve each equation. State which method you used.
2
29. 2x − 8x = 12
Page 16
9-5 Solving Quadratic Equations by Using the Quadratic Formula
The solution is 3.
28. AMUSEMENT PARKS The Demon Drop at Cedar Point in Ohio takes riders to the top of a tower and drops
2
them 60 feet. A function that approximates this ride is h = −16t + 64t − 60, where h is the height in feet and t is the
time in seconds. About how many seconds does it take for riders to drop 60 feet?
SOLUTION: 2
−16t + 64t − 60 = 0
For this equation, a = –16, b = 64, and c = –60.
It takes 2.5 seconds for the riders to drop from 60 feet to 0 feet.
Solve each equation. State which method you used.
2
29. 2x − 8x = 12
SOLUTION: Solve using the quadratic formula.
Write the equation in standard form.
For this equation, a = 2, b = –8, and c = –12.
Use the Quadratic Formula.
The solutions are 5.2 and –1.2.
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2
30. 3x − 24x = −36
SOLUTION: Page 17
9-5 Solving Quadratic Equations by Using the Quadratic Formula
It takes 2.5 seconds for the riders to drop from 60 feet to 0 feet.
Solve each equation. State which method you used.
2
29. 2x − 8x = 12
SOLUTION: Solve using the quadratic formula.
Write the equation in standard form.
For this equation, a = 2, b = –8, and c = –12.
Use the Quadratic Formula.
The solutions are 5.2 and –1.2.
2
30. 3x − 24x = −36
SOLUTION: Solve by using the quadratic formula.
Write the equation in standard form.
For this equation, a = 3, b = –24, and c = 36.
Use the Quadratic Formula.
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The solutions are 6 and 2.
2
Page 18
9-5 Solving
Quadratic
The solutions
are 5.2Equations
and –1.2. by Using the Quadratic Formula
2
30. 3x − 24x = −36
SOLUTION: Solve by using the quadratic formula.
Write the equation in standard form.
For this equation, a = 3, b = –24, and c = 36.
Use the Quadratic Formula.
The solutions are 6 and 2.
2
31. x − 3x = 10
SOLUTION: Solve by factoring.
Write the equation in standard form.
Factor.
The solutions are 5 and –2.
2
32. 4x + 100 = 0
SOLUTION: Solve using the quadratic formula.
First, check the value of the discriminant.
For this equation, a = 4, b = 0, and c = 100.
The discriminant is negative, so the equation has no real solutions, ø.
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2
33. x = −7x − 5
Page 19
9-5 Solving Quadratic Equations by Using the Quadratic Formula
The solutions are 5 and –2.
2
32. 4x + 100 = 0
SOLUTION: Solve using the quadratic formula.
First, check the value of the discriminant.
For this equation, a = 4, b = 0, and c = 100.
The discriminant is negative, so the equation has no real solutions, ø.
2
33. x = −7x − 5
SOLUTION: Solve by completing the square.
2
First, isolate the x- and the x -terms.
The solutions are –0.8 and –6.2.
34. 12 − 12x = −3x
2
SOLUTION: Solve by factoring.
Write the equation in standard form.
Factor.
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- Powered
The Manual
solution
is 2. by Cognero
Page 20
State the value of the discriminant for each equation. Then determine the number of real solutions of the
9-5 Solving
Quadratic Equations by Using the Quadratic Formula
The solutions are –0.8 and –6.2.
34. 12 − 12x = −3x
2
SOLUTION: Solve by factoring.
Write the equation in standard form.
Factor.
The solution is 2.
State the value of the discriminant for each equation. Then determine the number of real solutions of the
equation.
2
35. 0.2x − 1.5x + 2.9 = 0
SOLUTION: For this equation, a = 0.2, b = –1.5, and c = 2.9.
The discriminant is –0.07.
Since the discriminant is negative, the equation has no real solutions.
2
36. 2x − 5x + 20 = 0
SOLUTION: For this equation, a = 2, b = –5, and c = 20.
The discriminant is –135.
Since the discriminant is negative, the equation has no real solutions.
2
37. x −
x=3
SOLUTION: Write the equation in standard form.
For this equation, a = 1, b = –
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, and c = –3.
Page 21
9-5 Solving
Quadratic
Equations by Using the Quadratic Formula
The discriminant
is –135.
Since the discriminant is negative, the equation has no real solutions.
2
37. x −
x=3
SOLUTION: Write the equation in standard form.
For this equation, a = 1, b = –
, and c = –3.
The discriminant is 12.64.
Since the discriminant is positive, the equation has two real solutions.
2
38. 0.5x − 2x = −2
SOLUTION: Write the equation in standard form.
For this equation, a = 0.5, b = –2, and c = 2.
The discriminant is 0.
Since the discriminant is 0, the equation has one real solution.
2
39. 2.25x − 3x = −1
SOLUTION: Write the equation in standard form.
For this equation, a = 2.25, b = –3, and c = 1.
The discriminant is 0.
Since the discriminant is 0, the equation has one real solution.
40. SOLUTION: eSolutions
Manual - Powered by Cognero
Write the equation in standard form.
Page 22
9-5 Solving
Quadratic
Equations by Using the Quadratic Formula
The discriminant
is 0.
Since the discriminant is 0, the equation has one real solution.
40. SOLUTION: Write the equation in standard form.
For this equation, a = 2, b =
, and c =
.
The discriminant is 18.25.
Since the discriminant is positive, the equation has two real solutions.
2
41. CCSS MODELING The percent of U.S. households with high-speed Internet h can be estimated by h = −0.2n +
7.2n + 1.5, where n is the number of years since 1990.
a. Use the Quadratic Formula to determine when 20% of the population will have high-speed Internet.
b. Is a quadratic equation a good model for this information? Explain.
SOLUTION: a. Write the equation in standard form.
For this equation, a = –0.2, b = 7.2, and c = –18.5.
Since n is the number of years since 1990 add the solutions to 1990. 1990 + 3 = 1993; 1990 + 33 = 2023
b. A quadratic equation not a good model for this information. The parabola has a maximum at about 66, meaning
only 66% of the population would ever have high-speed Internet.
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Page 23
9-5 Solving
Quadratic
Equations by Using the Quadratic Formula
The discriminant
is 18.25.
Since the discriminant is positive, the equation has two real solutions.
2
41. CCSS MODELING The percent of U.S. households with high-speed Internet h can be estimated by h = −0.2n +
7.2n + 1.5, where n is the number of years since 1990.
a. Use the Quadratic Formula to determine when 20% of the population will have high-speed Internet.
b. Is a quadratic equation a good model for this information? Explain.
SOLUTION: a. Write the equation in standard form.
For this equation, a = –0.2, b = 7.2, and c = –18.5.
Since n is the number of years since 1990 add the solutions to 1990. 1990 + 3 = 1993; 1990 + 33 = 2023
b. A quadratic equation not a good model for this information. The parabola has a maximum at about 66, meaning
only 66% of the population would ever have high-speed Internet.
[-3. 48] scl: 5, by [-2, 68] scl: 7
2
42. TRAFFIC The equation d = 0.05v + 1.1v models the distance d in feet it takes a car traveling at a speed of v miles
per hour to come to a complete stop. If Hannah’s car stopped after 250 feet on a highway with a speed limit of 65
miles per hour, was she speeding? Explain your reasoning.
SOLUTION: Write the equation in standard form.
For this equation, a = 0.05, b = 1.1, and c = –250.
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Page 24
9-5 Solving Quadratic Equations by Using the Quadratic Formula
[-3. 48] scl: 5, by [-2, 68] scl: 7
2
42. TRAFFIC The equation d = 0.05v + 1.1v models the distance d in feet it takes a car traveling at a speed of v miles
per hour to come to a complete stop. If Hannah’s car stopped after 250 feet on a highway with a speed limit of 65
miles per hour, was she speeding? Explain your reasoning.
SOLUTION: Write the equation in standard form.
For this equation, a = 0.05, b = 1.1, and c = –250.
No, she was not speeding; Sample answer: Hannah was traveling at about 61 mph, so she was not speeding.
Without graphing, determine the number of x-intercepts of the graph of the related function for each
function.
43. 4.25x + 3 = −3x
2
SOLUTION: Write the equation in standard form.
For this equation, a = 3, b = 4.25, and c = 3.
The discriminant is –17.9.
Since the discriminant is negative, the graph of the function will not have any x-intercepts.
44. SOLUTION: Write the equation in standard form.
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For this equation, a = 1, b =
, and c =
.
Page 25
The discriminant is –17.9.
9-5 Solving Quadratic Equations by Using the Quadratic Formula
Since the discriminant is negative, the graph of the function will not have any x-intercepts.
44. SOLUTION: Write the equation in standard form.
For this equation, a = 1, b =
, and c =
.
The discriminant is 0.04.
Since the discriminant is positive, the graph of the function will have two x-intercepts.
2
45. 0.25x + x = −1
SOLUTION: Write the equation in standard form.
For this equation, a = 0.25, b = 1, and c = 1.
The discriminant is 0.
Since the discriminant is 0, the graph of the function will have one x-intercept.
Solve each equation by using the Quadratic Formula. Round to the nearest tenth if necessary.
2
46. −2x − 7x = −1.5
SOLUTION: Write the equation in standard form.
For this equation, a = –2, b = –7, and c = 1.5.
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Page 26
9-5 Solving
Quadratic
Equations by Using the Quadratic Formula
The discriminant
is 0.
Since the discriminant is 0, the graph of the function will have one x-intercept.
Solve each equation by using the Quadratic Formula. Round to the nearest tenth if necessary.
2
46. −2x − 7x = −1.5
SOLUTION: Write the equation in standard form.
For this equation, a = –2, b = –7, and c = 1.5.
The solutions are –3.7 and 0.2.
2
47. 2.3x − 1.4x = 6.8
SOLUTION: Write the equation in standard form.
For this equation, a = 2.3, b = –1.4, and c = –6.8.
The solutions are 2.1 and –1.4.
2
48. x − 2x = 5
SOLUTION: Write
the equation
eSolutions
Manual
- Powered in
by standard
Cognero form.
Page 27
9-5 Solving Quadratic Equations by Using the Quadratic Formula
The solutions are 2.1 and –1.4.
2
48. x − 2x = 5
SOLUTION: Write the equation in standard form.
For this equation, a = 1, b = –2, and c = –5.
The solutions are 3.4 and –1.4.
49. POSTER Bartolo is making a poster for the dance. He wants to cover three fourths of the area with text.
a. Write an equation for the area of the section with text.
b. Solve the equation by using the Quadratic Formula.
c. What should be the margins of the poster?
SOLUTION: a. The length of the area covered by text 20 – (x + x) or 20 – 2x. The width of the area is 25 – (4x + 3x) or 25 –
2
7x. The area of the entire poster is 20 in. × 25 in., or 500 in . Because Bartolo wants the area with text to be threefourths of the total area, it must be
ft. So, an equation for the area of the section with text is (20 − 2x)
(25 − 7x) = 375.
b. Write the equation in standard form.
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For this equation, a = 14, b = –190, and c = 125.
Page 28
9-5 Solving Quadratic Equations by Using the Quadratic Formula
The solutions are 3.4 and –1.4.
49. POSTER Bartolo is making a poster for the dance. He wants to cover three fourths of the area with text.
a. Write an equation for the area of the section with text.
b. Solve the equation by using the Quadratic Formula.
c. What should be the margins of the poster?
SOLUTION: a. The length of the area covered by text 20 – (x + x) or 20 – 2x. The width of the area is 25 – (4x + 3x) or 25 –
2
7x. The area of the entire poster is 20 in. × 25 in., or 500 in . Because Bartolo wants the area with text to be threefourths of the total area, it must be
ft. So, an equation for the area of the section with text is (20 − 2x)
(25 − 7x) = 375.
b. Write the equation in standard form.
For this equation, a = 14, b = –190, and c = 125.
The solutions are 12.9 and 0.7.
c. The margins should be about 0.7 in. on the sides, 4 • 0.7 = 2.8 in. on the top, and 3 • 0.7 = 2.1 in. on the bottom.
50. MULTIPLE REPRESENTATIONS In this problem, you will investigate writing a quadratic equation with given
roots. 2
2
If p is a root of 0 = ax + bx + c, then (x – p ) is a factor of ax + bx + c.
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Page 29
The solutions are 12.9 and 0.7.
c. The margins should be about 0.7 in. on the sides, 4 • 0.7 = 2.8 in. on the top, and 3 • 0.7 = 2.1 in. on the bottom.
50. MULTIPLE REPRESENTATIONS In this problem, you will investigate writing a quadratic equation with given
9-5 Solving Quadratic Equations by Using the Quadratic Formula
roots. 2
2
If p is a root of 0 = ax + bx + c, then (x – p ) is a factor of ax + bx + c.
a. Tabular Copy and complete the first two columns of the table.
b. Algebraic Multiply the factors to write each equation with integral coefficients. Use the equations to complete
the last column of the table. Write each equation.
c. Analytical How could you write an equation with three roots? Test your conjecture by writing an equation with
roots 1, 2, and 3. Is the equation quadratic? Explain.
SOLUTION: a. For any two roots m and p , in the left hand column, the middle column will be the corresponding factors (x – m), (x
– p ). b. The equation with these factors will be: (x – m)(x – p ) = 0 which simplifies to x2 – (m + p )x + mp = 0. Use this to
fill in the column of the table. c. You could write an equation with three roots by multiplying the corresponding factors together and setting it equal
to zero. If an equation has the three roots 1, 2, 3, then the corresponding factors would be (x – 1), (x – 2), and (x –
3). The equation would then be:
This is not a quadratic equation since it is of degree 3. eSolutions Manual - Powered by Cognero
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9-5 Solving Quadratic Equations by Using the Quadratic Formula
This is not a quadratic equation since it is of degree 3. 2
51. CHALLENGE Find all values of k such that 2x − 3x + 5k = 0 has two solutions.
SOLUTION: For the equation to have two solutions, the discriminant must be positive.
2
52. REASONING Use factoring techniques to determine the number of real zeros of f (x) = x − 8x + 16. Compare this
method to using the discriminant.
SOLUTION: 2
2
2
For f(x) = x − 8x + 16, a = 1, b = −8 and c = 16. Then the discriminate is b − 4ac or (−8) −4(1)(16) = 0. The 2.
polynomial can be factored to get f (x) = (x − 4) Solve to find the real zeros.
So the only real zero is 4. The discriminant is 0, so the only real zero is 4. The discriminant is 0, so there is 1 real
zero. The discriminant tells us how many real zeros there are. Factoring tells us what they are.
CCSS STRUCTURE Determine whether there are two, one, or no real solutions.
53. The graph of a quadratic function does not have an x-intercept.
SOLUTION: If there are no x-intercepts, then there are no real solutions.
54. The graph of a quadratic function touches but does not cross the x-axis.
SOLUTION: If the graph is tangent to the x-axis, meaning there is only one x-intercept, then there is only one real solution.
55. The graph of a quadratic function intersects the x-axis twice.
SOLUTION: If there are two x-intercepts, then there are two real solutions.
56. Both a and b are greater than 0 and c is less than 0 in a quadratic equation.
SOLUTION: eSolutions Manual - Powered by Cognero
2
2
Page 31
The discrimininant is b – 4ac. No matter the value of b, b will always be positive. If a is greater than 0 and c is
less than 0, then – 4ac will be positive. Thus the discrimininant would be positive. So there would be two real solutions.
If the graph is tangent to the x-axis, meaning there is only one x-intercept, then there is only one real solution.
55. The graph of a quadratic function intersects the x-axis twice.
SOLUTION: 9-5 Solving
Quadratic Equations by Using the Quadratic Formula
If there are two x-intercepts, then there are two real solutions.
56. Both a and b are greater than 0 and c is less than 0 in a quadratic equation.
SOLUTION: 2
2
The discrimininant is b – 4ac. No matter the value of b, b will always be positive. If a is greater than 0 and c is
less than 0, then – 4ac will be positive. Thus the discrimininant would be positive. So there would be two real solutions.
57. WRITING IN MATH Why can the discriminant be used to confirm the number of real solutions of a quadratic
equation?
SOLUTION: Consider a few examples:
,
For the first equation we have a = 1, b = 1, and c = 6.
.
In this case the discriminant is negative and the roots are imaginary. For the second equation we have: a = 1, b = 1, and c = –6.
When the discriminant is positive, the roots are real. We can only have imaginary roots if we take the square root of
a negative number, which will only happen if the discriminant is negative. 58. WRITING IN MATH Describe the advantages and disadvantages of each method of solving quadratic equations.
Which method do you prefer, and why?
SOLUTION: Factoring:
Factoring is easy if the polynomial is factorable and complicated if it is not. Not all equations are factorable. 2
2
2
For example f (x) = x – 8x + 16 factors to (x – 4) . However, f (x) = x – 16x + 8 can not be factored.
eSolutions Manual - Powered by Cognero
Page 32
Graphing:
Graphing only gives approximate answers, but it is easy to see the number of solutions. Using square roots is easy
58. WRITING IN MATH Describe the advantages and disadvantages of each method of solving quadratic equations.
Which method do you prefer, and why?
SOLUTION: 9-5 Solving
Quadratic Equations by Using the Quadratic Formula
Factoring:
Factoring is easy if the polynomial is factorable and complicated if it is not. Not all equations are factorable. 2
2
2
For example f (x) = x – 8x + 16 factors to (x – 4) . However, f (x) = x – 16x + 8 can not be factored.
Graphing:
Graphing only gives approximate answers, but it is easy to see the number of solutions. Using square roots is easy
when there is no x-term. 2
For example, for the quadratic f (x) = 2x – 17x + 4, you can see the two solutions in the graph. However, it will be
difficult to identify the solution x = 8.2578049 in the graph. [-5, 15] scl: 2 by [-30, 10] scl: 4
Completing the square:
Completing the square can be used for any quadratic equation and exact solutions can be found, but the leading
2
coefficient has to be 1 and the x - and x-term must be isolated. It is also easier if the coefficient of the x-term is
2
even; if not, the calculations become harder when dealing with fractions. For example x + 4x = 7 can be solved by
completing the square. Quadratic Formula:
The Quadratic Formula will work for any quadratic equation and exact solutions can be found. This method can be
time consuming, especially if an equation is easily factored. For example, use the Quadratic Formula to find the 2
solutions of f (x) = 4x + 13 x + 5. eSolutions Manual - Powered by Cognero
Page 33
Quadratic Formula:
The Quadratic Formula will work for any quadratic equation and exact solutions can be found. This method can be
9-5 Solving Quadratic Equations by Using the Quadratic Formula
time consuming, especially if an equation is easily factored. For example, use the Quadratic Formula to find the 2
solutions of f (x) = 4x + 13 x + 5. See students’ preferences.
59. If n is an even integer, which expression represents the product of three consecutive even integers?
A n(n + 1)(n + 2)
B (n + 1)(n + 2)(n + 3)
C 3n + 2
D n(n + 2)(n + 4)
SOLUTION: Adding 2 to an even integer results in the next consecutive integer.
n
n +2
n + 2 + 2, or n + 4
So, the correct choice is D.
60. SHORT RESPONSE The triangle shown is an isosceles triangle. What is the value of x?
SOLUTION: Because an isosceles triangle has two equal angles, x could be equal to 64, or it could be equal to the unnamed angle,
where 180 = 2x + 64.
The value of x is 58 or 64.
61. Which statement best describes the graph of x = 5?
F It is parallel to the x-axis.
G It is parallel to the y-axis.
H It passes through the point (2, 5).
J It Manual
has a y-intercept
5.
eSolutions
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Cognero
SOLUTION: A line for which x is always 5, is a vertical line, which is parallel to the y-axis.
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9-5 Solving Quadratic Equations by Using the Quadratic Formula
The value of x is 58 or 64.
61. Which statement best describes the graph of x = 5?
F It is parallel to the x-axis.
G It is parallel to the y-axis.
H It passes through the point (2, 5).
J It has a y-intercept of 5.
SOLUTION: A line for which x is always 5, is a vertical line, which is parallel to the y-axis.
So, the correct choice is G.
2
62. What are the solutions of the quadratic equation 6h + 6h = 72?
A 3 or −4
B −3 or 4
C no solution
D 12 or −48
SOLUTION: Write the equation in standard form.
For this equation, a = 6, b = 6, and c = –72.
The solutions are –4 or 3.
So, the correct choicer is A.
Solve each equation by completing the square. Round to the nearest tenth if necessary.
2
63. 6x − 17x + 12 = 0
SOLUTION: eSolutions Manual - Powered by Cognero
Page 35
The solutions
are –4Equations
or 3.
9-5 Solving
Quadratic
by Using the Quadratic Formula
So, the correct choicer is A.
Solve each equation by completing the square. Round to the nearest tenth if necessary.
2
63. 6x − 17x + 12 = 0
SOLUTION: The solutions are
or .
2
64. x − 9x = −12
SOLUTION: The solutions are 1.6 or 7.4.
2
65. 4x = 20x − 25
SOLUTION: eSolutions Manual - Powered by Cognero
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9-5 Solving Quadratic Equations by Using the Quadratic Formula
The solutions are 1.6 or 7.4.
2
65. 4x = 20x − 25
SOLUTION: Describe the transformations needed to obtain the graph of g(x) from the graph of f (x).
2
66. f (x) = 4x
2
g(x) = 2x
SOLUTION: 2
2
The graph of g(x) = ax stretches or compresses the graph of f (x) = 4x vertically. The change in a is
< 1. If 0 < 2
, and 0 <
2
< 1, the graph of f (x) = x is compressed vertically. Therefore, the graph of y = 2x is the graph of
2
y = 4x vertically compressed.
2
67. f (x) = x + 5
2
g(x) = x − 1
SOLUTION: 2
The graph of f (x) = x + c represents a vertical translation of the parent graph. The value of the change in c is –6,
2
and –6 < 0. If c < 0, the graph of f (x) = x is translated
2
units down. Therefore, the graph of y = x – 1 is a
2
translation of the graph of y = x +5 shifted down 6 units.
2
68. f (x) = x − 6
2
g(x) = x + 3
SOLUTION: 2
The graph of f (x) = x + c represents a vertical translation of the parent graph. The value of the change in c is 9, and
2
2
9 > 0. If c > 0, the graph of f (x) = x is translated units up. Therefore, the graph of y = x +3 is a translation of
2
the graph
y = x by
shifted
–6Cognero
eSolutions
Manualof
- Powered
up 9 units.
Page 37
Determine whether each graph shows a positive correlation , a negative correlation, or no correlation. If
there is a positive or negative correlation, describe its meaning in the situation.
SOLUTION: 2
The graph of f (x) = x + c represents a vertical translation of the parent graph. The value of the change in c is –6,
2
2
and –6 < 0. If c < 0, the graph of f (x) = x is translated units down. Therefore, the graph of y = x – 1 is a
9-5 Solving Quadratic Equations by Using the Quadratic Formula
2
translation of the graph of y = x +5 shifted down 6 units.
2
68. f (x) = x − 6
2
g(x) = x + 3
SOLUTION: 2
The graph of f (x) = x + c represents a vertical translation of the parent graph. The value of the change in c is 9, and
2
2
9 > 0. If c > 0, the graph of f (x) = x is translated units up. Therefore, the graph of y = x +3 is a translation of
2
the graph of y = x –6 shifted up 9 units.
Determine whether each graph shows a positive correlation , a negative correlation, or no correlation. If
there is a positive or negative correlation, describe its meaning in the situation.
69. SOLUTION: The graph shows a positive correlation. As time goes on, more people use electronic tax returns.
70. SOLUTION: The graph shows no correlation between the year and the number of hurricanes because the points are randomly
spread out.
71. ENTERTAINMENT Coach Washington wants to take her softball team out for pizza and soft drinks after the last
game of the season. A large pizza costs $12, and a pitcher of a soft drink costs $3. She does not want to spend more
than $60. Write an inequality that represents this situation, and graph the solution set.
SOLUTION: Let x = the number of pizzas she can buy and y = the number of pitchers of soft drink she can buy.
The graph crosses the y-axis at (0, 20) and has a slope of –4. Use a solid line because the inequality is ≤.
Test the point (0, 0) to see which side of the graph to shade.
Since
0 is less
than by
or Cognero
equal to
eSolutions
Manual
- Powered
60, shade the half-plane that contains (0, 0).
Page 38
70. SOLUTION: The graph
shows noEquations
correlationby
between
and the Formula
number of hurricanes because the points are randomly
9-5 Solving
Quadratic
Using the
theyear
Quadratic
spread out.
71. ENTERTAINMENT Coach Washington wants to take her softball team out for pizza and soft drinks after the last
game of the season. A large pizza costs $12, and a pitcher of a soft drink costs $3. She does not want to spend more
than $60. Write an inequality that represents this situation, and graph the solution set.
SOLUTION: Let x = the number of pizzas she can buy and y = the number of pitchers of soft drink she can buy.
The graph crosses the y-axis at (0, 20) and has a slope of –4. Use a solid line because the inequality is ≤.
Test the point (0, 0) to see which side of the graph to shade.
Since 0 is less than or equal to 60, shade the half-plane that contains (0, 0).
Determine whether each sequence is arithmetic, geometric, or neither. Explain.
72. 20, 25, 30, ...
SOLUTION: Check the difference and ratio between terms. 25 – 20 = 5
30 – 25 = 5
There is a common difference of 5 between the terms. The sequence is arithmetic.
73. 1000, 950, 900, ...
SOLUTION: Check the difference and ratio between terms. 950 – 1000 = –50
900 – 950 = –50 There is a common difference of –50 between the terms. The sequence is arithmetic.
74. 200, 350, 650, …
SOLUTION: Check the difference and ratio between terms. eSolutions
Manual - Powered by Cognero
350 – 200 = 150
650 – 350 = 200
Page 39
950 – 1000 = –50
900 – 950 = –50 9-5 Solving
Quadratic Equations by Using the Quadratic Formula
There is a common difference of –50 between the terms. The sequence is arithmetic.
74. 200, 350, 650, …
SOLUTION: Check the difference and ratio between terms. 350 – 200 = 150
650 – 350 = 200
There is no common difference between the terms. There is no common ratio. This is neither an arithmetic nor geometric sequence. 75. 6, 18, 54, …
SOLUTION: Check the difference and ratio between terms.
18 – 6 = 12
54 – 18 = 36
There is no common difference between terms. The common ratio between terms is 3. This is a geometric sequence. 76. 2, 4, 16, …
SOLUTION: Check the difference and ratio between terms. 4–2=2
16 – 4 = 12
There is no common difference between the terms. There is no common difference or ratio between the terms. This is neither a geometric nor an arithmetic sequence. 77. 8, –4, 2 …
SOLUTION: Check the difference and ratio between terms. eSolutions Manual - Powered by Cognero
–4 – 8 = –12
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9-5 Solving
Quadratic Equations by Using the Quadratic Formula
There is no common difference or ratio between the terms. This is neither a geometric nor an arithmetic sequence. 77. 8, –4, 2 …
SOLUTION: Check the difference and ratio between terms. –4 – 8 = –12
2 – (–4) = 6
There is no common difference between the terms.
The common ratio between the terms is
eSolutions Manual - Powered by Cognero
. This is a geometric sequence. Page 41