Review Exercises

Extracting Square Roots

Solve by extracting the roots.

1. $x^2-16=0$

2. $y^2=\frac{9}{4}$

3. $x^2-27=0$

4. $x^2+27=0$

5. $3y^2-25=0$

6. $9x^2-2=0$

7. ${\left(x-5\right)}^2-9=0$

8. ${\left(2x-1\right)}^2-1=0$

9. $16{\left(x-6\right)}^2-3=0$

10. $2{\left(x+3\right)}^2-5=0$

11. $\left(x+3\right)\left(x-2\right)=x+12$

12. $\left(x+2\right)\left(5x-1\right)=9x-1$

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

13. $\pm \sqrt{2}$

14. $\pm 2\sqrt{5}$

Completing the Square

Complete the square.

15. $x^2-6x+?={\left(x-?\right)}^2$

16. $x^2-x+?={\left(x-?\right)}^2$

Solve by completing the square.

17. $x^2-12x+1=0$

18. $x^2+8x+3=0$

19. $y^2-4y-14=0$

20. $y^2-2y-74=0$

21. $x^2+5x-1=0$

22. $x^2-7x-2=0$

23. $2x^2+x-3=0$

24. $5x^2+9x-2=0$

25. $2x^2-16x+5=0$

26. $3x^2-6x+1=0$

27. $2y^2+10y+1=0$

28. $5y^2+y-3=0$

29. $x\left(x+9\right)=5x+8$

30. $\left(2x+5\right)\left(x+2\right)=8x+7$

Quadratic Formula

Identify the coefficients a, b, and c used in the quadratic formula. Do not solve.

31. $x^2-x+4=0$

32. $-x^2+5x-14=0$

33. $x^2-5=0$

34. $6x^2+x=0$

Use the quadratic formula to solve the following.

35. $x^2-6x+6=0$

36. $x^2+10x+23=0$

37. $3y^2-y-1=0$

38. $2y^2-3y+5=0$

39. $5x^2-36=0$

40. $7x^2+2x=0$

41. $-x^2+5x+1=0$

42. $-4x^2-2x+1=0$

43. $t^2-12t-288=0$

44. $t^2-44t+484=0$

45. ${\left(x-3\right)}^2-2x=47$

46. $9x\left(x+1\right)-5=3x$

Guidelines for Solving Quadratic Equations and Applications

Use the discriminant to determine the number and type of solutions.

47. $-x^2+5x+1=0$

48. $-x^2+x-1=0$

49. $4x^2-4x+1=0$

50. $9x^2-4=0$

Solve using any method.

51. $x^2+4x-60=0$

52. $9x^2+7x=0$

53. $25t^2-1=0$

54. $t^2+16=0$

55. $x^2-x-3=0$

56. $9x^2+12x+1=0$

57. $4{\left(x-1\right)}^2-27=0$

58. ${\left(3x+5\right)}^2-4=0$

59. $\left(x-2\right)\left(x+3\right)=6$

60. $x\left(x-5\right)=12$

61. $\left(x+1\right)\left(x-8\right)+28=3x$

62. $\left(9x-2\right)\left(x+4\right)=28x-9$

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 $V\left(t\right)=125t^2-\mathrm{3,000}t+\mathrm{22,000}$, 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 $h\left(t\right)=-16t^2+48t$, 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. $y=2x^2+5x-3$

68. $y=x^2-12$

69. $y=5x^2-x+2$

70. $y=-x^2+10x-25$

Find the vertex and the line of symmetry.

71. $y=x^2-6x+1$

72. $y=-x^2+8x-1$

73. $y=x^2+3x-1$

74. $y=9x^2-1$

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

75. $y=x^2+8x+12$

76. $y=-x^2-6x+7$

77. $y=-2x^2-4$

78. $y=x^2+4x$

79. $y=4x^2-4x+1$

80. $y=-2x^2$

81. $y=-2x^2+8x-7$

82. $y=3x^2-1$

Determine the maximum or minimum y-value.

83. $y=x^2-10x+1$

84. $y=-x^2+12x-1$

85. $y=-5x^2+6x$

86. $y=2x^2-x-1$

87. The value in dollars of a new car is modeled by the function $V\left(t\right)=125t^2-\mathrm{3,000}t+\mathrm{22,000}$, 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 $h\left(t\right)=-16t^2+48t$, 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. $\sqrt{-36}$

90. $\sqrt{-40}$

91. $\sqrt{-\frac{8}{25}}$

92. $-\sqrt{-\frac{1}{9}}$

Perform the operations.

93. $\left(2-5i\right)+\left(3+4i\right)$

94. $\left(6-7i\right)-\left(12-3i\right)$

95. $\left(2-3i\right)\left(5+i\right)$

96. $\frac{4-i}{2-3i}$

Solve.

97. $9x^2+25=0$

98. $3x^2+1=0$

99. $y^2-y+5=0$

100. $y^2+2y+4$

101. $4x\left(x+2\right)+5=8x$

102. $2\left(x+2\right)\left(x+3\right)=3\left(x^2+13\right)$

Sample Exam

Solve by extracting the roots.

1. $4x^2-9=0$

2. ${\left(4x+1\right)}^2-5=0$

Solve by completing the square.

3. $x^2+10x+19=0$

4. $x^2-x-1=0$

Solve using the quadratic formula.

5. $-2x^2+x+3=0$

6. $x^2+6x-31=0$

Solve using any method.

7. $\left(5x+1\right)\left(x+1\right)=1$

8. $\left(x+5\right)\left(x-5\right)=65$

9. $x\left(x+3\right)=-2$

10. $2{\left(x-2\right)}^2-6=3x^2$

Set up an algebraic equation and solve.

11. The length of a rectangle is twice its width. If the diagonal measures $6\sqrt{5}$ 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 $h(t)=-16t^2+256t+3$, 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. $y=2x^2-4x-6$

14. $y=-x^2+4x-4$

15. $y=4x^2-9$

16. $y=x^2+2x-1$

17. Determine the maximum or minimum y-value: $y=-3x^2+12x-15$.

18. Determine the x- and y-intercepts: $y=x^2+x+4$.

19. Determine the domain and range: $y=25x^2-10x+1$.

20. The height in feet reached by a model rocket launched from a platform is given by the function $h(t)=-16t^2+256t+3$, 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 $R=200n-n^2$, 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: $\sqrt{-60}$.

23. Divide: $\frac{4-2i}{4+2i}$.

Solve.

24. $25x^2+3=0$

25. $-2x^2+5x-1=0$

Review Exercises Answers

1: ±16

3: $\pm 3\sqrt{3}$

5: $\pm \frac{5\sqrt{3}}{3}$

7: 2, 8

9: $\frac{24\pm \sqrt{3}}{4}$

11: $\pm 3\sqrt{2}$

13: $x^2-2=0$

15: $x^2-6x+9={\left(x-3\right)}^2$

17: $6\pm \sqrt{35}$

19: $2\pm 3\sqrt{2}$

21: $\frac{-5\pm \sqrt{29}}{2}$

23: −3/2, 1

25: $\frac{8\pm 3\sqrt{6}}{2}$

27: $\frac{-5\pm \sqrt{23}}{2}$

29: $-2\pm 2\sqrt{3}$

31: $a=1$, $b=-1$, and $c=4$

33: $a=1$, $b=0$, and $c=-5$

35: $3\pm \sqrt{3}$

37: $\frac{1\pm \sqrt{13}}{6}$

39: $\pm \frac{6\sqrt{5}}{5}$

41: $\frac{5\pm \sqrt{29}}{2}$

43: −12, 24

45: $4\pm 3\sqrt{6}$

47: Two real solutions

49: One real solution

51: −10, 6

53: ±1/5

55: $\frac{1\pm \sqrt{13}}{2}$

57: $\frac{2\pm 3\sqrt{3}}{2}$

59: −4, 3

61: $5\pm \sqrt{5}$

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: $x=-\frac{3}{2}$

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: $\frac{2i\sqrt{2}}{5}$

93: $5-i$

95: $13-13i$

97: $\pm \frac{5i}{3}$

99: $\frac{1}{2}\pm \frac{\sqrt{19}}{2}i$

101: $\pm \frac{i\sqrt{5}}{2}$

Sample Exam Answers

1: $\pm \frac{3}{2}$

3: $-5\pm \sqrt{6}$

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: $\left[0,\infty \right)$

21: To maximize revenue, the company needs to produce and sell 100 bicycles a week.

23: $\frac{3}{5}-\frac{4}{5}i$

25: $\frac{5\pm \sqrt{17}}{4}$