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9.7 Review Exercises and Sample Exam

Review Exercises

Extracting Square Roots

Solve by extracting the roots.

1. x216=0

2. y2=94

3. x227=0

4. x2+27=0

5. 3y225=0

6. 9x22=0

7. (x5)29=0

8. (2x1)21=0

9. 16(x6)23=0

10. 2(x+3)25=0

11. (x+3)(x2)=x+12

12. (x+2)(5x1)=9x1

Find a quadratic equation in standard form with the given solutions.

13. ±2

14. ±25

Completing the Square

Complete the square.

15. x26x+?=(x?)2

16. x2x+?=(x?)2

Solve by completing the square.

17. x212x+1=0

18. x2+8x+3=0

19. y24y14=0

20. y22y74=0

21. x2+5x1=0

22. x27x2=0

23. 2x2+x3=0

24. 5x2+9x2=0

25. 2x216x+5=0

26. 3x26x+1=0

27. 2y2+10y+1=0

28. 5y2+y3=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. x2x+4=0

32. x2+5x14=0

33. x25=0

34. 6x2+x=0

Use the quadratic formula to solve the following.

35. x26x+6=0

36. x2+10x+23=0

37. 3y2y1=0

38. 2y23y+5=0

39. 5x236=0

40. 7x2+2x=0

41. x2+5x+1=0

42. 4x22x+1=0

43. t212t288=0

44. t244t+484=0

45. (x3)22x=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+x1=0

49. 4x24x+1=0

50. 9x24=0

Solve using any method.

51. x2+4x60=0

52. 9x2+7x=0

53. 25t21=0

54. t2+16=0

55. x2x3=0

56. 9x2+12x+1=0

57. 4(x1)227=0

58. (3x+5)24=0

59. (x2)(x+3)=6

60. x(x5)=12

61. (x+1)(x8)+28=3x

62. (9x2)(x+4)=28x9

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)=125t23,000t+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+5x3

68. y=x212

69. y=5x2x+2

70. y=x2+10x25

Find the vertex and the line of symmetry.

71. y=x26x+1

72. y=x2+8x1

73. y=x2+3x1

74. y=9x21

Graph. Find the vertex and the y-intercept. In addition, find the x-intercepts if they exist.

75. y=x2+8x+12

76. y=x26x+7

77. y=2x24

78. y=x2+4x

79. y=4x24x+1

80. y=2x2

81. y=2x2+8x7

82. y=3x21

Determine the maximum or minimum y-value.

83. y=x210x+1

84. y=x2+12x1

85. y=5x2+6x

86. y=2x2x1

87. The value in dollars of a new car is modeled by the function V(t)=125t23,000t+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. (25i)+(3+4i)

94. (67i)(123i)

95. (23i)(5+i)

96. 4i23i

Solve.

97. 9x2+25=0

98. 3x2+1=0

99. y2y+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. 4x29=0

2. (4x+1)25=0

Solve by completing the square.

3. x2+10x+19=0

4. x2x1=0

Solve using the quadratic formula.

5. 2x2+x+3=0

6. x2+6x31=0

Solve using any method.

7. (5x+1)(x+1)=1

8. (x+5)(x5)=65

9. x(x+3)=2

10. 2(x2)26=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=2x24x6

14. y=x2+4x4

15. y=4x29

16. y=x2+2x1

17. Determine the maximum or minimum y-value: y=3x2+12x15.

18. Determine the x- and y-intercepts: y=x2+x+4.

19. Determine the domain and range: y=25x210x+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=200nn2, 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: 42i4+2i.

Solve.

24. 25x2+3=0

25. 2x2+5x1=0

Review Exercises Answers

1: ±16

3: ±33

5: ±533

7: 2, 8

9: 24±34

11: ±32

13: x22=0

15: x26x+9=(x3)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: 5i

95: 1313i

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: 3545i

25: 5±174