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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. $3{y}^{2}-25=0$

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

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. $2{x}^{2}+x-3=0$

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

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

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

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

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

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. $6{x}^{2}+x=0$

*Use the quadratic formula to solve the following.*

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

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

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

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

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

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

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

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

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. $4{x}^{2}-4x+1=0$

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

*Solve using any method.*

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

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

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

54. ${t}^{2}+16=0$

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

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

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

68. $y={x}^{2}-12$

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

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. $y=9{x}^{2}-1$

*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. $y=-2{x}^{2}-4$

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

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

80. $y=-2{x}^{2}$

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

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

*Determine the maximum or minimum **y**-value.*

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

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

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

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

87. The value in dollars of a new car is modeled by the function $V\left(t\right)=125{t}^{2}-\mathrm{3,000}t+\mathrm{22,000}$, 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 $h\left(t\right)=-16{t}^{2}+48t$, 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. $9{x}^{2}+25=0$

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

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

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

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

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

*Solve by extracting the roots.*

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

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. $-2{x}^{2}+x+3=0$

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

*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 $h(t)=-16{t}^{2}+256t+3$, 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. $y=2{x}^{2}-4x-6$

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

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

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

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

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

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

20. The height in feet reached by a model rocket launched from a platform is given by the function $h(t)=-16{t}^{2}+256t+3$, 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. $25{x}^{2}+3=0$

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

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

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

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

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