# Quadratic Equations Chapter Questions

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1 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? NJ Center for Teaching and Learning ~ 1 ~

2 Quadratic Equations Chapter Questions Standard form and the axis of symmetry Find the axis of symmetry and rewrite in standard form if necessary. 1. y= x 2 +3x x 2 + 5x = 6 3. y= x 2-4x x 2-3 =6x 5. y= 3x 2-4x -2 Homework Find the axis of symmetry and rewrite in standard form if necessary. 6. y= x 2 +2x x 2-3x = 2 8. y= x 2-5x x - 4 = 2x y= 3x 2-2x Transforming Quadratic Functions Does the graph of the given equation open up or down? Is the graph wider or narrower than the parent equation of y=x 2? What is the y-intercept? 11. y = 2x 2 +3x y = -.7x 2-4x y = -1.2x y = 3x 2 +3x 15. y = -4x 2 Homework Does the graph of the given equation open up or down? Is the graph wider or narrower than the parent equation of y=x 2? What is the y-intercept? 16. y =-.6x 2 +3x y = 1.7x 2-4x y = -1.02x y = 1.3x 2 +4x 20. y = 5x 2 NJ Center for Teaching and Learning ~ 2 ~

3 Graphing to Find Zeros Find the zeros of the following quadratics: 21. y x y x y = x 2-4x y = -x 2 +3x y = -x 2-8x y = -x 2-8x y = x 2 +3x y = 2x 2 +4x y = -3x 2 +4x +4 NJ Center for Teaching and Learning ~ 3 ~

4 Homework 30. y x y x y = x 2-6x y = -x 2 +3x y = x 2 +6x y = x 2 +x y = x 2 +2x y = 2x 2 +5x y = -3x 2 +11x +4 Solving by Factoring Solve the following quadratics by factoring. 39. a 2 +4a +3= b 2-4b -5= c 2-6c = d 2 +8c = -12 NJ Center for Teaching and Learning ~ 4 ~

5 43. -e 2 +9 = f 2 +4f +4 = g 2 +5g = h 2 +7h +6= j 2-4j = 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. a 2 +6a +5= b 2 -b -6= c 2-6c = d 2 +7c = e = f 2 +6f +9 = g 2 +7g = h 2 +8h +6= j 2-7j = 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 Solve the following quadratics using the square roots method 59. m 2 = n 2 = p 2 = q 2 = r 2-3 =6 64. s 2 +8 = t 2-6 = u 2 +5 = (v -7) 2-5 = (w -3) 2 +6 = 56 NJ Center for Teaching and Learning ~ 5 ~

6 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. m 2 = n 2 = p 2 = q 2 = r 2-3 = s 2 +8 = t 2-6 = u 2 +5 = (v -2) 2 +4 = (w +4) 2-4 = 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 Fill in the blank to complete the square. 81. a 2 + 8a b b c 2-4c d 2-6d e 2 + 7e f 2-5f + Solve the following quadratics by completing the square. 87. h 2 + 6h = j 2-8j = k = -10k 90. m 2-13 = 12m n + 20 = -n p + p 2 = q 2-8q = r r = 12 NJ Center for Teaching and Learning ~ 6 ~

7 95. A toy rocket launched into the air has a height (h feet) at any given time (t seconds) as h = -16t t 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. a a b 2 + b c 2-14c d 2-16d e 2 + 9e f 2-1f + Solve the following quadratics by completing the square h 2 + 4h = j 2-10j = k = -14k 105. m 2-21 = 20m n + 80= -n p + p 2 = q 2-12q = r r = A toy rocket launched into the air has a height (h feet) at any given time (t seconds) as h = -16t t 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 Solve the following using the quadratic formula. Round answers to the hundredth place x 2 +8x -6 = g 2-4g +2 = d 2 + 4d -3 = m 2 + 3m = w 2-8 = 5w z 9z 2 = 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) NJ Center for Teaching and Learning ~ 7 ~

8 Homework Solve the following using the quadratic formula. Round answers to the hundredth place x 2 +7x -5 = g 2-5g +3 = d 2 + 5d -3 = m 2 + 4m = w 2-2 = 5w z 6z 2 = 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 Find the discriminant for each quadratic equation. State the number of real roots and then find the real solution(s), if any exist x 2-6x + 5 = x 2-4x - 6 = x + 4x = x 9 = x x 2 = 6x x 2-9x 2= x (2x 5) = A rock is thrown with a height equation of h = -16t t + 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 x 2-9x + 7 = x 2-4x + 2 = x + 7x = x 6 = 2x x 2 = 7x 6 NJ Center for Teaching and Learning ~ 8 ~

9 138. 3x 2-7x 8= (x + 3)(2x + 6) = A rock is thrown with a height equation of h = -16t t + 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 Solve the following problems using any method The product of two consecutive positive integers is 272, find the integers The product of two consecutive positive even integers is 528, find the integers The product of two consecutive odd integers is 255, find the integers 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 The product of two consecutive positive integers is 272, find the integers The product of two consecutive positive even integers is 342, find the integers The product of two consecutive odd integers is 483, find the integers 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 square feet what was the perimeter of the original square? NJ Center for Teaching and Learning ~ 9 ~

10 154. 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 = 5x 2 + 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 = -3x 2-12x 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 = x 2 + 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 = -2x 2 + 2x 3? A) (0, 5) B) B (3, 0) C) C (0, -3) D) D (-3, 0) NJ Center for Teaching and Learning ~ 10 ~

11 5.Which graph(s) has more than one zero? 6.Which of the following is a step in solving y = 2x 2 - 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: x 2-6x +? A) -9 B) -36 C) 36 D) 9 NJ Center for Teaching and Learning ~ 11 ~

12 9.What is the discriminant of 2x 2 + 6x + 2 = 0? A) 2 B) 6 C) 20 D) 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 3x 2 + 5x + 1 =0 A) B) C) D) ± Simplify 2 A) -8 and -2 B) -3 and 1 C) -3 and -1 D) 3 and 2 NJ Center for Teaching and Learning ~ 12 ~

13 13. Given the height of a rocket as h = -16t t + 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 3x 2 + 7x + 4 = 0 A) - 1 and 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: x 2 + x + 36? 16. Solve 6x x + 6 = 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? NJ Center for Teaching and Learning ~ 13 ~

14 Quadratic Equations CW HW Answer Key 1. axis of sym. = - 3/2 2. axis of sym. =5/2; y = x 2-5x axis of sym. =2 4. axis of sym. =- 3/2; y= 2x 2 + 6x axis of sym = 2/3 6. axis of sym. = axis of sym. =- 3/2;y = x 2 +3x axis of sym. =5/2; 9. axis of sym. = - 5/4; y = 2x 2 + 5x axis of sym. =1/3 11. Up ;narrower; Down; wider; Down;narrower; Up; narrower; Down; narrower; Down; wider; Up; narrower; Down; narrower; Up; narrower; Up; narrower; Zeros = 1 and Zeros = 1 and Zeros = 1 and Zeros = none 25. Zeros = -3 and Zeros = Zeros = -5 and Zeros = Zeros = 2 and -2/3 30. Zeros = 1 and Zeros = 3 and Zeros = 1 and Zeros = 5 and Zeros = Zeros = 3 and Zeros = none 37. Zeros = -2 and Zeros = - 1/3 and and and and and and and /2 and /3 and Length = 8 ft and and and and and and and and 4/3 58. Length = 4 ft. 59. ±4 60. ±5 61. ±2 62. ±4 63. ±3 64. ±3 65. ±1 66. ± and and (x 6) 2 = 25 x = 1 or m=±6 NJ Center for Teaching and Learning ~ 14 ~

15 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 w=0 or (x + 5) 2 = 72 x = 1 or h=2 or j = 1 or k = -9 or m = 13 or n 1.61 or p = 0 or q 6.9 or r.42 or t 5.93 sec h=2 or j=9 or k=-1 or m=21 or no solution 107. p=-6 or no solution 109. r=1 or t.06 sec. or 9.94 sec or or or m= 1 or 1/ or or hours or or or or or or hrs two solutions; x = 5 or two solutions; x = 3 or no solution 128. one solution; x= no solution 130. two solutions; 2.45 or two solutions; 3.81 or No, the max. ht. is about 11.2 ft no solutions 134. one solution ; x= no solutions 136. two solutions; x = 3/2 or no solutions 138. two solutions; 3.17 or two solutions; -.66 or Yes, max ht. is 69 ft & & & 17 NJ Center for Teaching and Learning ~ 15 ~

16 144. About mi About 61.4 mi ft ,000 sq ft & & & About 371 miles 152. About 70.7 miles ft ,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 and square feet 18. X = 7 and units; 55 ; 20 units 21 NJ Center for Teaching and Learning ~ 16 ~

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