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This is “Review Exercises and Sample Exam”, section 9.7 from the book Beginning Algebra (v. 1.0). For details on it (including licensing), click here.
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9.7 Review Exercises and Sample Exam
Review Exercises
Extracting Square Roots
Solve by extracting the roots.
1. x2−16=0
2. y2=94
3. x2−27=0
4. x2+27=0
5. 3y2−25=0
6. 9x2−2=0
7. (x−5)2−9=0
8. (2x−1)2−1=0
9. 16(x−6)2−3=0
10. 2(x+3)2−5=0
11. (x+3)(x−2)=x+12
12. (x+2)(5x−1)=9x−1
Find a quadratic equation in standard form with the given solutions.
13. ±2
14. ±25
Completing the Square
Complete the square.
15. x2−6x+?=(x−?)2
16. x2−x+?=(x−?)2
Solve by completing the square.
17. x2−12x+1=0
18. x2+8x+3=0
19. y2−4y−14=0
20. y2−2y−74=0
21. x2+5x−1=0
22. x2−7x−2=0
23. 2x2+x−3=0
24. 5x2+9x−2=0
25. 2x2−16x+5=0
26. 3x2−6x+1=0
27. 2y2+10y+1=0
28. 5y2+y−3=0
29. x(x+9)=5x+8
30. (2x+5)(x+2)=8x+7
Quadratic Formula
Identify the coefficients a, b, and c used in the quadratic formula. Do not solve.
31. x2−x+4=0
32. −x2+5x−14=0
33. x2−5=0
34. 6x2+x=0
Use the quadratic formula to solve the following.
35. x2−6x+6=0
36. x2+10x+23=0
37. 3y2−y−1=0
38. 2y2−3y+5=0
39. 5x2−36=0
40. 7x2+2x=0
41. −x2+5x+1=0
42. −4x2−2x+1=0
43. t2−12t−288=0
44. t2−44t+484=0
45. (x−3)2−2x=47
46. 9x(x+1)−5=3x
Guidelines for Solving Quadratic Equations and Applications
Use the discriminant to determine the number and type of solutions.
47. −x2+5x+1=0
48. −x2+x−1=0
49. 4x2−4x+1=0
50. 9x2−4=0
Solve using any method.
51. x2+4x−60=0
52. 9x2+7x=0
53. 25t2−1=0
54. t2+16=0
55. x2−x−3=0
56. 9x2+12x+1=0
57. 4(x−1)2−27=0
58. (3x+5)2−4=0
59. (x−2)(x+3)=6
60. x(x−5)=12
61. (x+1)(x−8)+28=3x
62. (9x−2)(x+4)=28x−9
Set up an algebraic equation and use it to solve the following.
63. The length of a rectangle is 2 inches less than twice the width. If the area measures 25 square inches, then find the dimensions of the rectangle. Round off to the nearest hundredth.
64. An 18-
65. The value in dollars of a new car is modeled by the function V(t)=125t2−3,000t+22,000, where t represents the number of years since it was purchased. Determine the age of the car when its value is $22,000.
66. The height in feet reached by a baseball tossed upward at a speed of 48 feet/second from the ground is given by the function h(t)=−16t2+48t, where t represents time in seconds. At what time will the baseball reach a height of 16 feet?
Graphing Parabolas
Determine the x-
67. y=2x2+5x−3
68. y=x2−12
69. y=5x2−x+2
70. y=−x2+10x−25
Find the vertex and the line of symmetry.
71. y=x2−6x+1
72. y=−x2+8x−1
73. y=x2+3x−1
74. y=9x2−1
Graph. Find the vertex and the y-
75. y=x2+8x+12
76. y=−x2−6x+7
77. y=−2x2−4
78. y=x2+4x
79. y=4x2−4x+1
80. y=−2x2
81. y=−2x2+8x−7
82. y=3x2−1
Determine the maximum or minimum y-
83. y=x2−10x+1
84. y=−x2+12x−1
85. y=−5x2+6x
86. y=2x2−x−1
87. The value in dollars of a new car is modeled by the function V(t)=125t2−3,000t+22,000, where t represents the number of years since it was purchased. Determine the age of the car when its value is at a minimum.
88. The height in feet reached by a baseball tossed upward at a speed of 48 feet/second from the ground is given by the function h(t)=−16t2+48t, where t represents time in seconds. What is the maximum height of the baseball?
Introduction to Complex Numbers and Complex Solutions
Rewrite in terms of i.
89. −36
90. −40
91. −825
92. −−19
Perform the operations.
93. (2−5i)+(3+4i)
94. (6−7i)−(12−3i)
95. (2−3i)(5+i)
96. 4−i2−3i
Solve.
97. 9x2+25=0
98. 3x2+1=0
99. y2−y+5=0
100. y2+2y+4
101. 4x(x+2)+5=8x
102. 2(x+2)(x+3)=3(x2+13)
Sample Exam
Solve by extracting the roots.
1. 4x2−9=0
2. (4x+1)2−5=0
Solve by completing the square.
3. x2+10x+19=0
4. x2−x−1=0
Solve using the quadratic formula.
5. −2x2+x+3=0
6. x2+6x−31=0
Solve using any method.
7. (5x+1)(x+1)=1
8. (x+5)(x−5)=65
9. x(x+3)=−2
10. 2(x−2)2−6=3x2
Set up an algebraic equation and solve.
11. The length of a rectangle is twice its width. If the diagonal measures 65 centimeters, then find the dimensions of the rectangle.
12. The height in feet reached by a model rocket launched from a platform is given by the function h(t)=−16t2+256t+3, where t represents time in seconds after launch. At what time will the rocket reach 451 feet?
Graph. Find the vertex and the y-
13. y=2x2−4x−6
14. y=−x2+4x−4
15. y=4x2−9
16. y=x2+2x−1
17. Determine the maximum or minimum y-
18. Determine the x-
19. Determine the domain and range: y=25x2−10x+1.
20. The height in feet reached by a model rocket launched from a platform is given by the function h(t)=−16t2+256t+3, where t represents time in seconds after launch. What is the maximum height attained by the rocket.
21. A bicycle manufacturing company has determined that the weekly revenue in dollars can be modeled by the formula R=200n−n2, where n represents the number of bicycles produced and sold. How many bicycles does the company have to produce and sell in order to maximize revenue?
22. Rewrite in terms of i: −60.
23. Divide: 4−2i4+2i.
Solve.
24. 25x2+3=0
25. −2x2+5x−1=0
Review Exercises Answers
1: ±16
3: ±33
5: ±533
7: 2, 8
9: 24±34
11: ±32
13: x2−2=0
15: x2−6x+9=(x−3)2
17: 6±35
19: 2±32
21: −5±292
23: −3/2, 1
25: 8±362
27: −5±232
29: −2±23
31: a=1, b=−1, and c=4
33: a=1, b=0, and c=−5
35: 3±3
37: 1±136
39: ±655
41: 5±292
43: −12, 24
45: 4±36
47: Two real solutions
49: One real solution
51: −10, 6
53: ±1/5
55: 1±132
57: 2±332
59: −4, 3
61: 5±5
63: Length: 6.14 inches; width: 4.07 inches
65: It is worth $22,000 new and when it is 24 years old.
67: x-
69: x-
71: Vertex: (3, −8); line of symmetry: x=3
73: Vertex: (−3/2, −13/4); line of symmetry: x=−32
75:
77:
79:
81:
83: Minimum: y = −24
85: Maximum: y = 9/5
87: The car will have a minimum value 12 years after it is purchased.
89: 6i
91: 2i25
93: 5−i
95: 13−13i
97: ±5i3
99: 12±192i
101: ±i52
Sample Exam Answers
1: ±32
3: −5±6
5: −1, 3/2
7: −6/5, 0
9: −2, −1
11: Length: 12 centimeters; width: 6 centimeters
13:
15:
17: Maximum: y = −3
19: Domain: R; range: [0,∞)
21: To maximize revenue, the company needs to produce and sell 100 bicycles a week.
23: 35−45i
25: 5±174
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