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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. $(2x+5)(x+2)=8x+7$

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-foot ladder leaning against a building reaches a height of 17 feet. How far is the base of the ladder from the wall? Round to the nearest tenth of a foot.

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- and y-intercepts. 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-intercept. In addition, find the x-intercepts if they exist. 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-value. 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-intercept. In addition, find the x-intercepts if they exist. 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-value: $y=−3x2+12x−15$. 18. Determine the x- and y-intercepts: $y=x2+x+4$. 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-intercepts: (−3, 0), (1/2, 0); y-intercept: (0, −3)

69: x-intercepts: none; y-intercept: (0, 2)

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: $±i52$

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: $5±174$