Quadratic Equations Chapter Questions - NJCTLcontent.njctl.org/courses/math/algebra-i/quadratic...Aug 31, 2013 · Quadratic Equations Chapter Questions Standard form and the axis

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Quadratic Equations Chapter Questions

1. Describe the characteristics of a quadratic equation.

2. What are the steps for graphing a quadratic function?

3. How can you determine the number of solutions without graphing?


Quadratic Equations Chapter Questions

Standard form and the axis of symmetry

Class Work

Find the axis of symmetry and rewrite in standard form if necessary.

1. y= x2 +3x - 4

2. -x 2 + 5x = 6

3. y= x2 -4x -2

4. -2x2 - 3 =6x

5. y= 3x2 -4x -2

Homework

Find the axis of symmetry and rewrite in standard form if necessary.

6. y= x2 +2x -8

7. -x2 - 3x = 2

8. y= x2 -5x -1

9. -5x - 4 = 2x2

10. y= 3x2 -2x

Transforming Quadratic Functions

Class Work

Does the graph of the given equation open up or down? Is the graph wider or narrower than the parent

equation of y=x2? What is the y-intercept?

11. y = 2x2 +3x -4

12. y = -.7x2 -4x +3

13. y = -1.2x2 +6

14. y = 3x2 +3x

15. y = -4x2

Homework

Does the graph of the given equation open up or down? Is the graph wider or narrower than the parent

equation of y=x2? What is the y-intercept?

16. y =-.6x2 +3x -6

17. y = 1.7x2 -4x +5

18. y = -1.02x2 +8

19. y = 1.3x2 +4x

20. y = 5x2


Graphing to Find Zeros

Class Work

Find the zeros of the following quadratics:

21.

22.

-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

-8

-6

-4

-2

2

4

6

8

x

y

-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

-8

-6

-4

-2

2

4

6

8

x

y

23. y = x2 -4x +3

24. y = -x2 +3x -8

25. y = -x2 -8x -15

26. y = -x2 -8x -16

27. y = x2 +3x -10

28. y = 2x2 +4x +2

29. y = -3x2 +4x +4


Homework

30.

31.

-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

-8

-6

-4

-2

2

4

6

8

x

y

-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

-8

-6

-4

-2

2

4

6

8

x

y

32. y = x2 -6x +5

33. y = -x2 +3x +10

34. y = x2 +6x +9

35. y = x2 +x -12

36. y = x2 +2x +4

37. y = 2x2 +5x +2

38. y = -3x2 +11x +4

Solving by Factoring

Class Work

Solve the following quadratics by factoring.

39. a2 +4a +3= 0

40. b2 -4b -5= 0

41. -c2 -6c = -7

42. d2 +8c = -12


43. -e2 +9 = 0

44. f2 +4f +4 = 0

45. –g2 +5g = 6

46. 2h2 +7h +6= 0

47. 3j2 -4j = -1

48. A garden has a length of (x + 2) feet and a width of (2x - 1) feet. The garden’s total area is 88 square

feet. Find the length.

Home Work

Solve the following quadratics by factoring.

49. a2 +6a +5= 0

50. b2 -b -6= 0

51. c2 -6c = -8

52. d2 +7c = -10

53. -e2 +16 = 0

54. f2 +6f +9 = 0

55. –g2 +7g = 6

56. 2h2 +8h +6= 0

57. 3j2 -7j = -4

58. A garden has a length of (x - 4)feet and a width of (2x +3)feet. The garden’s total area is 76 square

feet. Find the length.

Solving Using the Square Roots Method

Class Work

Solve the following quadratics using the square roots method

59. m2 = 16

60. n2 = 25

61. 3p2 = 12

62. 5q2 = 80

63. r2 -3 =6

64. s2 +8 =17

65. 2t2 -6 = -4

66. 3u2 +5 = 17

67. (v -7)2 -5 = 11

68. 2(w -3)2 +6 = 56


69. The square of six less than a number is twenty-five. Write an equation that models this situation.

Solve the equation.

Homework

Solve the following quadratics using the square roots method

70. m2 = 36

71. n2 = 64

72. 3p2 = 27

73. 5q2 = 20

74. r2 -3 = 13

75. s2 +8 = 24

76. 2t2 -6 = 12

77. 3u2 +5 = 8

78. (v -2)2 +4 = 13

79. 3(w +4)2 -4 = 44

80. Two times the square of five more than a number is seventy-two. Write an equation that models this

situation. Solve the equation.

Solving by Completing the Square

Class Work

Fill in the blank to complete the square.

81. a2 + 8a +__

82. b2 + 10b +__

83. c2 - 4c +__

84. d2 - 6d +__

85. e2 + 7e +__

86. f2 - 5f +__

Solve the following quadratics by completing the square.

87. h2 + 6h =16

88. j2 - 8j = -7

89. k2 + 9 = -10k

90. m2 - 13 = 12m

91. 14n + 20 = -n2

92. 8p + p2 = 0

93. 2q2 - 8q = 40

94. 3r2 + 36r = 12


95. A toy rocket launched into the air has a height (h feet) at any given time (t seconds) as h = -16t2 + 96t

until it hits the ground. At what time(s) is it at a height of 7 feet above the ground?

Homework

Fill in the blank to complete the square.

96. a2 + 12a +__

97. b2 + b +__

98. c2 - 14c +__

99. d2 - 16d +__

100. e2 + 9e +__

101. f2 - 1f +__

Solve the following quadratics by completing the square.

102. h2 + 4h =12

103. j2 - 10j = -9

104. k2 + 13 = -14k

105. m2 - 21 = 20m

106. 2n + 80= -n2

107. 6p + p2 = 0

108. 2q2 - 12q = -22

109. 3r2 + 15r = 18

110. A toy rocket launched into the air has a height (h feet) at any given time (t seconds) as

h = -16t2 + 160t until it hits the ground. At what time(s) is it at a height of 9 feet above the

ground?

Solving Using the Quadratic Formula

Class Work

Solve the following using the quadratic formula. Round answers to the hundredth place.

111. x2 +8x -6 =0

112. g2 -4g +2 =0

113. 3d2 + 4d -3 =0

114. -2m2 + 3m = 1

115. 4w2 -8 = 5w

116. 7z – 9z2 = -4

117. An employee makes (2x + 3) dollars an hour for x hours. If the employer wants to pay no more

than $120 a day, what is the maximum number of hours the employee can work? (Round to the

nearest quarter hour)


Homework

Solve the following using the quadratic formula. Round answers to the hundredth place.

118. x2 +7x -5 =0

119. g2 -5g +3 =0

120. 2d2 + 5d -3 =0

121. -3m2 + 4m = -5

122. 5w2 -2 = 5w

123. 3z – 6z2 = -8

124. An employee makes (3x - 5) dollars an hour for x hours. If the employer wants to pay no more

than $200 a day, what is the maximum number of hours the employee can work? (Round to the

nearest quarter hour)

Discriminant

Class Work

Find the discriminant for each quadratic equation. State the number of real roots and then find the real

solution(s), if any exist.

125. x2 - 6x + 5 = 0

126. 2x2 - 4x - 6 = 0

127. 3x + 4x2 + 5 = 0

128. 6x – 9 = x2

129. 3x2 = 6x – 8

130. 4x2 - 9x – 2= 0

131. x (2x – 5) = 10

132. A rock is thrown with a height equation of h = -16t2 + 20t + 5 (where h is the height of the rock in

feet at any given time of t in seconds). Will it reach a height of 30 feet? Explain your answer.

Homework

Find the discriminant for each quadratic equation. State the number of real roots and then find the real

solution(s), if any exist.

133. 3x2 - 9x + 7 = 0

134. 2x2 - 4x + 2 = 0

135. 5x + 7x2 + 8 = 0

136. 7x – 6 = 2x2

137. 5x2 = 7x – 6


138. 3x2 - 7x – 8= 0

139. (x + 3)(2x + 6) = 11

140. A rock is thrown with a height equation of h = -16t2 + 64t + 5 (where h is the height of the rock in

feet at any given time of t in seconds). Will it reach a height of 50 feet? Explain your answer.

Mixed Application Problems

Class Work

Solve the following problems using any method.

141. The product of two consecutive positive integers is 272, find the integers.

142. The product of two consecutive positive even integers is 528, find the integers.

143. The product of two consecutive odd integers is 255, find the integers.

144. Two planes leave airport at the same time (from different runways). If three hours later they are

500 miles apart and the plane flying south has traveled 200 miles farther, how far did the one

flying west travel?

145. Two cars leave a gas station at the same time, one traveling north and one traveling east. One

hour later they are 80 miles apart and the one traveling east went 10 miles farther, how far is it

from the gas station?

146. A square has its length increased by 4 feet and its width by 5 feet. If the resulting rectangle has

an area of 132 square feet what was the perimeter of the original square?

147. A rectangular parking lot has a width 30 feet more than its length. The owners are able to

increase the width by 20 feet and the length by 40. The new lot has an area of 27,200 square

feet, what is the area of the original lot?

Homework

Solve the following problems using any method.

148. The product of two consecutive positive integers is 272, find the integers.

149. The product of two consecutive positive even integers is 342, find the integers.

150. The product of two consecutive odd integers is 483, find the integers.

151. Two planes leave airport at the same time (from different runways). If three hours later they are

600 miles apart and the plane flying south has traveled 100 miles farther, how far did the one

flying west travel?

152. Two cars leave a gas station at the same time, one traveling north and one traveling east. One

hour later they are 90 miles apart and the one traveling east went 15 miles farther, how far is it

from the gas station?

153. A square has its length increased by 6 feet and its width by 8 feet. If the resulting rectangle has

an area of 239.25 square feet what was the perimeter of the original square?


154. A rectangular parking lot has a width 20 feet more than its length. The owners are able to

increase the width by 20 feet and the length by 40. The new lot has an area of 7225 square feet,

what is the area of the original lot?

Unit Review

Multiple Choice– Choose the correct answer for each question. No partial credit will be given.

1. Comparing the graph of y = 5x2 + 4x - 2 to its parent function, it:

A) opens down and is wider than the parent function graph.

B) opens down and is narrower than the parent function graph.

C) opens up and is wider than the parent function graph.

D) opens up and is narrower than the parent function graph.

2. What is the equation of the axis of symmetry of y = -3x2 - 12x – 5?

A) x = -2

B) x = -4

C) x = 4

D) x = 2

3. What are the vertex and axis of symmetry of the parabola

y = x2 + 4x + 3?

A) vertex: (2,−1); axis of symmetry: x = 2

B) vertex: (2,1); axis of symmetry: x = 2

C) vertex: (−2,−1); axis of symmetry: x = −2

D) vertex: (−2,1); axis of symmetry: x = −2

4.What is the y- intercept of y = -2x2 + 2x – 3?

A) (0 , 5)

B) B (3 , 0)

C) C (0 , -3)

D) D (-3, 0)


5.Which graph(s) has more than

one zero?

6.Which of the following is a step in solving y = 2x2 - x - 3 by the Factoring Method?

A) 2x + 1 = 0 or x + 3 = 0

B) 2x - 3 = 0 or x + 1 = 0

C) 2x + 3 = 0 or x - 1 = 0

D) 2x - 1 = 0 or x - 3 = 0

7.The solution to (x + 2)2 = 16 is

A) 14 and -18

B) -14 and 18

C) -6 and 2

D) -2 and 6

8.What value goes in the blank to complete the square: x2 - 6x + ___?

A) -9

B) -36

C) 36

D) 9


9.What is the discriminant of 2x2 + 6x + 2 = 0?

A) 2

B) 6

C) 20

D) 28

10. How many real zeros does an equation have if the discriminant is -4?

A) 0

B) 1

C) 2

D) Not enough information

11. Solve 3x2 + 5x + 1 =0

A) 5 13

6

B) 5 13

6

C) 5 13

6

D) 5 13

6

12. Simplify -2 ± √16

2

A) -8 and -2

B) -3 and 1

C) -3 and -1

D) 3 and 2


13. Given the height of a rocket as h = -16t2 + 160t + 896, where t is in seconds.

At what time t, does the rocket hit the ground?

A) - 14 and 4 seconds

B) 14 seconds

C) - 4 seconds

D) the rocket will not hit the ground

14. Solve 3x2 + 7x + 4 = 0

A) - 1 and - 4

3

B) 1 and 4

3

C) -7 and - 6

D) 7 and 6

Short Constructed Response – Write the correct answer for each question.

15. What value goes in the blank to complete the perfect square trinomial: x2 + __x + 36?

16. Solve 6x2 + 13x + 6 = 0

17. A square has its length increased by 4 feet and its width decreased by 3 feet; the resulting

area is 144 square feet. What was the area of the original square?

18. Solve (3x- 7)(x+3)=0

Extended Constructed Response - Solve the problem, showing all work.

19. A rectangle has a length 6 more than its width. If the width is decreased by 2 and the length

decreased by 4, the resulting rectangle has an area of 21 square units. What is the length of the

original rectangle? What is ratio of the original rectangle’s area to the new rectangle’s area?

What is the perimeter of the new rectangle?


Quadratic Equations CW HW Answer Key

1. axis of sym. = - 3/2

2. axis of sym. =5/2; y = x2 - 5x + 6

3. axis of sym. =2

4. axis of sym. =- 3/2; y= 2x2 + 6x + 3

5. axis of sym = 2/3

6. axis of sym. = - 1

7. axis of sym. =- 3/2;y = x2 +3x + 2

8. axis of sym. =5/2;

9. axis of sym. = - 5/4; y = 2x2 + 5x + 4

10. axis of sym. =1/3

11. Up ;narrower; -4

12. Down; wider; 3

13. Down;narrower; 6

14. Up; narrower; 0

15. Down; narrower; 0

16. Down; wider; -6

17. Up; narrower; 5

18. Down; narrower; 8

19. Up; narrower; 0

20. Up; narrower; 0

21. Zeros = 1 and - 5

22. Zeros = 1 and 3

23. Zeros = 1 and 3

24. Zeros = none

25. Zeros = -3 and -5

26. Zeros = - 4

27. Zeros = -5 and 2

28. Zeros = -1

29. Zeros = 2 and -2/3

30. Zeros = 1 and -6


32. Zeros = 1 and 5


34. Zeros = -3


36. Zeros = none

37. Zeros = -2 and -0.5

38. Zeros = - 1/3 and 4

39. -3 and -1

40. 5 and -1

41. 1 and -7

42. -6 and -2

43. -3 and 3

44. -2

45. 2 and 3

46. - 3/2 and -2

47. 1/3 and 1

48. Length = 8 ft.

49. -1 and -5

50. -2 and 3

51. 2 and 4

52. -2 and -5

53. 4 and -4

54. -3

55. 1 and 6

56. -1 and -3

57. 1 and 4/3

58. Length = 4 ft.

59. ±4

60. ±5

61. ±2

62. ±4

63. ±3

64. ±3

65. ±1

66. ±2

67. 3 and 11

68. 8 and -2

69. (𝑥 − 6)2 = 25 𝑥 = 1 𝑜𝑟 11

70. m=±6


71. n=±8

72. p=±3

73. q=±2

74. r=±4

75. s=±4

76. t=±3

77. u=±1

78. v=5 or -1

79. w=0 or -8

80. 2(𝑥 + 5)2 = 72 𝑥 = 1 𝑜𝑟 − 11

81. 16

82. 25

83. 4

84. 9

85. 12.25

86. 6.25

87. h=2 or -8

88. j = 1 or 7

89. k = -9 or -1

90. m = 13 or -1

91. n≈ −1.61 𝑜𝑟 12.39

92. p = 0 or -8

93. 𝑞 ≈ 6.9 𝑜𝑟 − 2.9

94. 𝑟 ≈ .42 𝑜𝑟 − 9.42

95. 𝑡 ≈ 5.93 𝑠𝑒𝑐.

96. 36

97. 1

98. 49

99. 64

100. 20.25

101. .25

102. h=2 or -6

103. j=9 or 1

104. k=-1 or -13

105. m=21 or -1

106. no solution

107. p=-6 or0

108. no solution

109. r=1 or -6

110. 𝑡 ≈ .06 𝑠𝑒𝑐. 𝑜𝑟 9.94 𝑠𝑒𝑐.

111. .69 or -8.69

112. 3.42 or .58

113. .54 or -1.87

114. m= 1 or 1/2

115. 2.17 𝑜𝑟 − .92

116. −.38 𝑜𝑟 1.16

117. 7 hours

118. .65 or -7.65

119. 4.3 𝑜𝑟 .7

120. 1

2 or

−9

4

121. -.79 or 2.12

122. 1.31 or -.31

123. -.93 or 1.43

124. 9 hrs.

125. two solutions; x = 5 or 1

126. two solutions; x = 3 or -1

127. no solution

128. one solution; x=3

129. no solution

130. two solutions; 2.45 or -.2

131. two solutions; 3.81 or -1.31

132. No, the max. ht. is about 11.2 ft.

133. no solutions

134. one solution ; x=1

135. no solutions

136. two solutions; x = 3/2 or 2

137. no solutions

138. two solutions; 3.17 or -.84

139. two solutions; -.66 or -5.35

140. Yes, max ht. is 69 ft.

141. 16 & 17

142. 22 & 24

143. 15 & 17


144. About 239.1 mi.

145. About 61.4 mi.

146. 28 ft.

147. 18,000 sq ft.

148. 19 & 20

149. 24 & 26

150. 21 & 23

151. About 371 miles

152. About 70.7 miles

153. 34 ft

154. 2,925 sq ft.

Review Answer Key

1. D

2. A

3. C

4. C

5. B, D

6. B

7. C

8. D

9. C

10. A

11. C

12. B

13. B

14. A

15. 12

16. −2

3𝑎𝑛𝑑

−3

2

17. 144 square feet

18. X = 7

3 and -3

19. 11 units; 55

21 ; 20 units

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Quadratic Equations Chapter Questions - NJCTLcontent.njctl.org/courses/math/algebra-i/quadratic...Aug 31, 2013 · Quadratic Equations Chapter Questions Standard form and the axis