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3 APPLY ASSIGNMENT GUIDE BASIC Day 1: p. 536 Exs. 24–52 even Day 2: pp. 537–538 Exs. 54–80 even, 84, 87–90, 92–102 even AVERAGE Day 1: p. 536 Exs. 24–52 even Day 2: pp. 537–538 Exs. 54–80 even, 82–84, 87–90, 92–102 even ADVANCED Day 1: p. 536 Exs. 24–52 even Day 2: pp. 537–538 Exs. 54–80 even, 82–90, 92–102 even BLOCK SCHEDULE pp. 536–538 Exs. 24–80 even, 82–84, 87–90, 92–102 even EXERCISE LEVELS Level A: Easier 87–91 Level B: More Difficult 23–84, 92–103 Level C: Most Difficult 85, 86 HOMEWORK CHECK To quickly check student understanding of key concepts, go over the following exercises: Exs. 24, 32, 40, 48, 54, 64, 70, 72, 78. See also the Daily Homework Quiz: • Blackline Master (Chapter 2 Resource Book, p. 83) • Transparency (p. 72) COMMON ERROR EXERCISE 44 Students may look at the equation and quickly assume that c is 30 instead of –30. Remind them to write a quadratic equation in standard form before using the quadratic formula. It may help students also to note that they can multiply through by –1 to remove the negative signs. 36–43. See Additional Answers beginning on page AA1. 536 GUIDED PRACTICE ✓ Concept Check ✓ Vocabulary Check Skill Check ✓ 1. What formula can you use to solve any quadratic equation? the quadratic formula 2. Explain how to use the quadratic formula to solve º2x 2 + 5x = º7. See margin. 3. What is the difference between the two models for vertical motion? See margin. Use the quadratic formula to solve the equation. 2. Sample answer: Rewrite 4. x 2 + 6x º 7 = 0 1, º7 5. x 2 º 2x º 15 = 0 5, º3 6. x 2 + 12x + 36 = 0 º6 the equation in standard form. Then substitute 7. 4x 2 º 8x + 3 = 0 8. 3x 2 + x º 1 = 0 9. x 2 + 6x º 3 = 0 a = º2, b = 5, and c = 7 0.5, 1.5 See margin. º3 + 2兹3苶, º3 º 2兹3苶 into the quadratic formula. Write in standard form. Use the quadratic formula to solve the equation. The solutions are 10. 2x 2 = –x + 6 11. 6x = –8x 2 + 2 12. 3 = 3x 2 + 8x º5 + 兹5苶2苶 º苶4(º 苶2) 苶(7 苶)苶 1 3 1 }}} , 2 3x 2 + 8x º 3 = 0; º3, }} 2(º2) } } } } 2x 2 + x º 6 = 0; º2, 8x + 6x º 2 = 0; º1, 3 2 2 14. ºx 2 + 4x = 3 4 15. 4x 2 + 4x = º1 13. º14x = –2x + 36 or x = º1, and 1 2 2 2 } } 2x º 14x º 36 = 0; º2, 9 ºx + 4x º 3 = 0; 1, 3 4x + 4x + 1 = 0; º 2 º5 º 兹5苶2苶 º苶4(º 苶2) 苶(7 苶)苶 Find the x-intercepts of the graph of the equation. }}} , or 2(º2) 16. y = x 2 º 11x + 24 17. y = x 2 + 10x + 16 18. y = 2x 2 + 4x º 30 3, 8 º2, º8 º5, 3 3. In one model, the object is 19. y = 2x 2 + 6x º 9 20. y = 5x 2 + 8x º 8 21. y = 4x 2 + 8x º 1 dropped, that is, its initial See margin. See margin. See margin. FIELD TARGET EVENT From a hot-air balloon directly over a target, you velocity is 0. In the other, 22. the object is thrown, and, throw a marker with an initial downward velocity of º40 feet per second from therefore, has a nonzero a height of 180 feet. How long does it take the marker to reach the target? 2.33 sec initial velocity. º3 º 3兹3苶 º4 º 2兹14 苶 º2 º 兹5苶 21. }} ≈ º4.10, 20. }} ≈ º2.30, ≈ º2.12, 19. }} 2 5 2 º1 + 兹13 苶 º1 º 兹13 苶 8. }}, }} º3 + 3兹3苶 º4 + 2兹14 苶 º2 + 兹5苶 6 6 }} ≈ 1.10 }} ≈ 0.70 }} ≈ 0.12 x = 3.5. 2 5 2 PRACTICE AND APPLICATIONS STUDENT HELP Extra Practice to help you master skills is on p. 805. FINDING VALUES Find the value of b 2 º 4ac for the equation. 23. x 2 º 3x º 4 = 0 25 24. 4x 2 + 5x + 1 = 0 9 26. 5w 2 º 3w º 2 = 0 49 27. s 2 º 13s + 42 = 0 1 1 29. 5x 2 + 5x + ᎏᎏ = 0 21 5 25. x 2 º 11x + 30 = 0 1 28. 3x 2 º 5x º 12 = 0 169 1 1 30. ᎏᎏa 2 + 5a º 8 = 0 41 31. ᎏᎏv 2 º 6v º 3 = 0 39 2 4 SOLVING EQUATIONS Use the quadratic formula to solve the equation. 1 46. x 2 º 11x + 16 = 0; 11 º 兹57 苶 }} ≈ 1.73, 2 11 + 兹57 苶 }} ≈ 9.27 2 STUDENT HELP HOMEWORK HELP Example 1: Example 2: Example 3: Example 4: Exs. 23–43 Exs. 44–52 Exs. 53–61 Exs. 71–80 32. 4x 2 º 13x + 3 = 0 }4}, 3 33. y 2 + 11y + 10 = 0 34. 7x 2 + 8x + 1 = 0 1 º10, º1 º1, ºᎏᎏ 35. º3y 2 + 2y + 8 = 0 4 36. 6n2 º 10n + 3 = 0 37. 9x 2 + 14x + 3 = 0 7 See margin. º}}, 2 See margin. 3 38. 8m2 + 6m º 1 = 0 39. 7x 2 + 2x º 1 = 0 40. º6x 2 º 3x + 2 = 0 See margin. See margin. See margin. 1 41. ºᎏᎏx 2 + 6x + 13 = 0 42. º10a 2 + 3a + 2 = 0 43. º2d 2 º 5d + 19 = 0 2 See margin. See margin. See margin. STANDARD FORM Write the quadratic equation in standard form. Solve using the quadratic formula. 44. 2x 2 = 4x + 30 45. 2x 2 º 4x º 30 = 0; º3, 5 2 47. 5x º 1 = º6x 1 48. 6x 2 + 5x º 1 = 0; º1, }} 6 51. 2 50. º1 + 3x = 2x 1 2 3x º 2x º 1 = 0; º}}, 1 3 536 2 º 3x + x 2 = 0 46. 16 = ºx 2 + 11x See margin. 2q 2 º 6 = –4q 49. 5z º 2z2 + 15 = 8 7 2q + 4q º 6 = 0; 1, º3 2z 2 º 5z º 7 = 0; º1, }} º5c2 + 9c = 4 52. º16b = º8b 2 º 8 2 4 8b 2 º16b + 8 = 0; 1 5c 2 º 9c + 4 = 0; }}, 1 x 2 2º 3x + 2 = 0; 1, 2 Chapter 9 Quadratic Equations and Functions 5 3 º 3 3苶 2 兹 56. }} ≈ º1.10, FINDING INTERCEPTS Find the x-intercepts of the graph of the equation. 3 + 3兹3苶 }} ≈ 4.10 2 º1 º 兹17 苶 ≈ º2.56, 57. }} 2 º7 º 57 苶 2 53. y = 3x 2 º 6x º 24 54. y = 2x 2 º 6x º 8 55. y = 2x 2 º 2x º 12 4, º2 4, º1 º2, 3 56. y = º2x 2 + 6x + 9 57. y = x 2 + x º 4 58. y = x 2 + 7x º 2 See margin. See margin. See margin. 59. y = º3x 2 º 2x + 1 1 60. y = º4x 2 + 8x º 2 61. y = º5x 2 + 5x + 5 º1, }} See margin. See margin. 3 CHOOSING A METHOD Solve the quadratic equation by finding square roots or by using the quadratic formula. Explain why you chose the method. 62–70. See margin. 62. 6x 2 + 20x + 5 = 0 63. m2 = 32 64. x 2 º 625 = 0 º7 + 兹57 苶 }} ≈ 0.27 2 65. 4y 2 º 49 = 0 66. º2x 2 + 6x + 1 = 0 67. h2 + 18h + 81 = 0 2 º 2苶 2 68. 5x 2 = 25 69. 9y 2 º 3y = 1 70. v 2 = 8v + 2 2 + 兹2苶 }} ≈ 1.71 2 FIELD TARGET EVENT In Exercises 71–76, six balloonists compete in the Field Target Event at a hot-air balloon festival. Calculate the amount of time it takes for the marker to hit the target when thrown down from the given initial height (in feet) with the given initial velocity (in feet per second). º1 + 兹17 苶 }} ≈ 1.56 2 兹 58. }} ≈ º7.27, 兹 60. }} ≈ 0.29, 1 º 兹5苶 ≈ º0.62, 61. }} 2 1 + 兹5苶 }} ≈ 1.62 2 71. s = 200; v = º50 72. s = 150; v = º25 73. s = 100; v = º10 2.30 sec 2.38 sec 2.21 sec 74. s = 150; v = º33 75. s = 80; v = º40 76. s = 50; v = 0 2.20 sec 1.31 sec 1.77 sec VERTICAL MOTION In Exercises 77–80, use a vertical motion model to find how long it will take for the object to reach the ground. 77. You drop keys from a window 30 feet above ground to your friend below. Your friend does not catch them. 1.37 sec 78. An acorn falls 45 feet from the top of a tree. 1.68 sec FOCUS ON 80. You throw a ball downward with an initial speed of 10 feet per second out of a window to a friend 20 feet below. Your friend does not catch the ball. 0.85 sec 81. 82. RE FE L AL I URBAN BIRDS Cities provide a habitat for many species of wildlife including birds of prey such as Peregrine falcons and red-tailed hawks. 83. PEREGRINE FALCON A falcon dives toward a pigeon on the ground. When the falcon is at a height of 100 feet the pigeon sees the falcon, which is diving at 220 feet per second. Estimate the time the pigeon has to escape. 0.44 sec Falcon 100 ft RED-TAILED HAWK A hawk dives toward a snake. When the hawk is at a height of 200 feet the snake sees the hawk, which is diving at 105 feet per second. Estimate the time the snake has to escape. 1.54 sec BALD EAGLE An eagle dives 62–70. Sample explanations are given. –10 + 兹70 苶 –10 – 兹70 苶 62. } ≈ –0.27, } ≈ 6 6 –3.06; quadratic formula; the expression on the left cannot be written as a perfect square. 63. 4兹2苶 ≈ 5.66, –4兹2苶 ≈ –5.66; finding square roots; simplest method 64. –25, 25; finding square roots; simplest method 7 7 65. – }}, }}; finding square roots; 2 2 simplest method –3 + 兹11 苶 79. A lacrosse player throws a ball upward from her playing stick with an initial height of 7 feet, at an initial speed of 90 feet per second. 5.70 sec APPLICATIONS APPLICATION NOTE EXERCISE 81 It is important to specify that the wings are folded so that air resistance is reduced and the dive is more like free fall. Students may have seen films in which sky divers fold in their arms to make a faster descent. Ski jumpers widen their arms and skis so that they fall less quickly. –3 – 兹11 苶 66. } ≈ –0.16, } ≈ 3.16; –2 –2 quadratic formula; the expression on the left cannot be written as a perfect square. 67. –9; finding square roots; the expression on the left can be written as (h + 9) 2. 68. 兹5苶 ≈ 2.24, –兹5苶 ≈ –2.24; finding square roots; simplest method 69 and 70. See Additional Answers beginning on page AA1. ADDITIONAL PRACTICE AND RETEACHING For Lesson 9.5: Pigeon toward a mouse. When the eagle is at a height of 125 feet the mouse sees the eagle, which is diving at 145 feet per second. Estimate the time the mouse has to escape. 0.79 sec 9.5 Solving Quadratic Equations by the Quadratic Formula 537 • Practice Levels A, B, and C (Chapter 9 Resource Book, p. 73) • Reteaching with Practice (Chapter 9 Resource Book, p. 76) • See Lesson 9.5 of the Personal Student Tutor For more Mixed Review: • Search the Test and Practice Generator for key words or specific lessons. 537 Test Preparation 4 ASSESS DAILY HOMEWORK QUIZ Transparency Available 1. Find the value of b 2 – 4ac for –2x 2 + 7x – 6. 1 2. Use the quadratic formula to solve 6x 2 – x – 15. – }32}, }53} 3. Write –17x = –6x 2 – 12 in standard form. Solve using the quadratic formula. 84. MULTI-STEP PROBLEM In parts (a)–(d), a batter hits a pitched baseball when it is 3 feet off the ground. After it is hit, the height h (in feet) of the ball at time t (in seconds) is modeled by 84. d. Sample answer: The speed of the ball approaching the h = º16t 2 + 80t + 3 batter, the speed of the swinging bat, the angle of where t is the time (in seconds). the bat, and the wind and a. Find the time when the ball hits the ground in the outfield. 5.04 sec weather conditions can change the path of the b. Write a quadratic equation that you can use to find the time when the baseball. The dimensions of baseball is at its maximum height of 103 feet. Solve the quadratic the ball park also determine equation. º16t 2 + 80t + 3 = 103, or º16t 2 + 80t º 100 = 0; 2.5 sec whether a hit is a home run. c. 4 3 3 2 6x 2 – 17x + 12 = 0; }}, }} 4. Find the x-intercepts of the graph of y = 2x 2 – 5x + 1. ★ Challenge 5 – 兹17 苶 5 + 兹17 苶 } ≈ 0.22, } ≈ 2.28 4 4 5. Solve 9x 2 – 36 = 0 by finding square roots or by using the quadratic formula. –2, 2 6. The initial upward velocity of a golf ball is 60 ft/sec. How long will it take the ball to return to the ground? 3.75 sec 1 2 85. x = 1}}; the x-intercepts are 冉 12 冊 EXTRA CHALLENGE www.mcdougallittell.com EXTRA CHALLENGE NOTE Challenge problems for Lesson 9.5 are available in blackline format in the Chapter 9 Resource Book, p. 80 and at www.mcdougallittell.com. ADDITIONAL TEST PREPARATION 1. WRITING Describe how to use the x-intercepts of the graph of a quadratic function to find the vertex of the graph. Sample answer: Find the x-coordinate of the point midway between the x-intercepts. This is the x-coordinate of the vertex. Substitute this value into the equation of the function to find the y-coordinate of the vertex. 92. 93. 4 5 6 ⫺2 ⫺1 0 1 2 3 4 538 0 2 4 6 94–102. See Additional Answers beginning on page AA1. 538 x Writing Explain how you can use this two-part form of the quadratic formula 兹b苶苶º 苶苶4a苶c苶 x = ᎏᎏ ± ᎏᎏ ºb 2a 2 2a to find the distance between the axis of symmetry of a parabola and either of its x-intercepts. See margin. ⫺21 Ex. 85 EVALUATING EXPRESSIONS Evaluate the expression. (Review 1.3 for 9.6) the axis of symmetry of the 2 88. ºy 2 when y = º1 º1 graph of y = ax 2 + bx + c. 87. x when x = º5 25 The given values of x are 89. º4xy when x = º2 and y = º6º48 90. y 2 º y when y = º2 6 the x-intercepts of the equation, which are NEQUALITIES Solve the inequality and graph the solution. (Review 6.3, 6.4) equidistant from the axis I91–96. See margin for graphs. of symmetry. The distance 91. 2 ≤ x < 5 92. 8 > 2x > º4 93. º12 < 2x º 6 < 4 of each point from the axis 4 > x > º2 º3 < x < 5 of symmetry is 94. º3 < ºx < 1 95. |x + 5| ≥ 10 96. |2x + 9| ≤ 15 º1 < x < 3 x ≤ º15 or x ≥ 5 º12 ≤ x ≤ 3 苶2苶苶 苶c苶 兹b º苶 4a SKETCHING GRAPHS Sketch the graph of the function. Label the vertex. }}. 2a (Review 9.3) 97–102. See margin for graphs. 1 5 97. }}, º}} 3 3 97. y = 6x 2 º 4x º 1 98. y = º3x 2 º 5x + 3 99. y = º2x 2 º 3x + 2 5 61 98. º}}, }} 6 12 1 1 100. y = ᎏᎏx 2 + 2x º 1 101. y = 4x 2 º ᎏᎏx + 4 102. y = º5x 2 º 0.5x + 0.5 3 25 2º3) 4 99. º}}, }} (º2, 4 8 103. RECREATION There were 1.4 ª 107 people who visited Golden Gate 1 255 101. }}, 3}} Recreation Area in California in 1996. On average, how many people visited 3 ⫺2 86. 3 ºb 2a 2 ⫺4 y 3 symmetry of the graph and show that it lies halfway between the two x-intercepts. See margin. 86. x = }} is the equation of 1 0 85. VISUAL THINKING Write an equation of the axis of MIXED REVIEW 冉 冊 冉 冊 冉 冊 冉 32 256 冊 1 41 102. 冉º}}, }}冊 20 80 91. 2.5 sec; this is the same as the value found in part (b). d. CRITICAL THINKING What factors change the path of a baseball? What factors would contribute to hitting a home run? See margin. º3 and 6, and the midpoint of (º3, 0) and (6, 0) is 1}}, 0 . Use a graphing calculator to graph the function. Use the zoom feature to approximate the time when the baseball is at its maximum height. Compare your results with those you obtained in part (b). per day? per month? 䉴 Source: National Park Service (Review 8.4) about 3.84 ª 10 4 people; about 1.17 ª 10 6 people Chapter 9 Quadratic Equations and Functions