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GUIDED PRACTICE Vocabulary Check ✓ Concept Check ✓ 3. Sample answer: Subtract c from both sides, then divide both sides by a. c If º}} is positive, find both a c c square roots of º}}. If º}} a a is zero, the only solution c is 0. If º}} is negative, a there are no real solutions. Skill Check ✓ 3 APPLY 1. State the meanings of the symbols 兹苶, º兹苶, and ± 兹苶. the principal (or positive) square root; the negative square root; both the positive and the negative square root 2. Give an example of a perfect square and an example of an irrational number. 苶 Sample answers: 121; 兹11 3. Explain how to find solutions of an equation of the form ax 2 + c = 0. See margin. If the statement is true, give an example. If false, give a counterexample. 4. Some numbers have no real square root. True. Sample answer: 兹º 苶1苶 5. No number has only one square root. False. Sample answer: 兹0苶 = 0 AVERAGE 6. All positive numbers have two different square roots. True. Sample answer: º5 and 5 are both square roots of 25. 7. The square root of the sum of two numbers is equal to the sum of the square roots of the numbers. False. Sample answer: 兹6苶+ 苶苶 3 = 3; 兹6苶 + 兹3苶 = 4.8 12. 9. 兹0 苶.8 苶1苶 6 0.9 10. º兹0 苶.0 苶4苶 º0.2 11. ±兹9 苶 Day 1: pp. 507–510 Exs. 23, 26, 29, and every 3rd Ex. through Ex. 77, 78–81, 87, 95–100, 104, 107, 110, 111 ±3 兹b苶2苶+ 苶苶10苶a苶 10 ± 2兹b苶 13. ᎏᎏ a 6 3, 7 14. 兹b苶2苶º 苶苶8a苶 BLOCK SCHEDULE 0 Solve the equation. If there is no solution, state the reason. 15. 2x 2 º 8 = 0 16. x 2 + 25 = 0 17. x 2 º 1.44 = 0 18. 5x 2 = º15 See margin. See margin. º2, 2 º1.2, 1.2 FALLING OBJECTS If an object is dropped from an initial height s, how long will it take to reach the ground? Assume there is no air resistance. 19. s = 48 about 1.7 sec 20. s = 96 about 2.4 sec 21. s = 192 about 3.5 sec 22. If you double the height from which the object falls, do you double the falling time? If not, why? No. Sample answer: The height is a function of the square of the falling time rather than of the falling time. PRACTICE AND APPLICATIONS STUDENT HELP Extra Practice to help you master skills is on p. 805. FINDING SQUARE ROOTS Find all square roots of the number or write no square roots. Check the results by squaring each root. 23. 49 º7, 7 27. 0 0 24. 144 º12, 12 28. 100 º10, 10 25. º4 no square roots 29. 0.09 º0.3, 0.3 26. º25 no square roots 30. 0.16 º0.4, 0.4 EVALUATING SQUARE ROOTS Evaluate the expression. Give the exact value if possible. Otherwise, approximate to the nearest hundredth. STUDENT HELP 31. º兹1 苶6苶9苶 º13 32. 兹6 苶4苶 HOMEWORK HELP 35. 兹0 苶.0 苶4苶 0.2 36. ±兹0 苶.7 苶5苶 Example 1: Exs. 23–30 Example 2: Exs. 31–38 Example 3: Exs. 39–44 continued on p. 508 Day 1: pp. 507–510 Exs. 23, 26, 29, and every 3rd Ex. through Ex. 77, 79–81, 87, 95, 96, 104, 107, 110, 111 ADVANCED Evaluate the expression. Evaluate the radical expression when a = 2 and b = 4. 18. There is no solution; negative numbers have no real square roots. BASIC Day 1: pp. 507–510 Exs. 23, 26, 29, and every 3rd Ex. through Ex. 77, 79–81, 95, 96, 104, 107, 110, 111 8. 兹3 苶6苶 16. There is no solution; negative numbers have no real square roots. ASSIGNMENT GUIDE 8 33. 兹1 苶3苶 3.61 ±0.87 37. º兹0 苶.1 苶 34. º兹1 苶2苶5苶 º11.18 º0.32 38. ±兹6 苶.2 苶5苶 ±2.5 EVALUATING EXPRESSIONS Evaluate 兹b 苶2苶º 苶苶4a苶c苶 for the given values. 39. a = 4, b = 5, c = 1 3 42. a = 2, b = 4, c = 0.5 2兹3苶 40. a = º2, b = 8, c = º8 0 43. a = º3, b = 7, c = 5 兹10 苶9苶 41. a = 3, b = º7, c = 6 undefined 44. a = 6, b = º8, c = 4 undefined 9.1 Solving Quadratic Equations by Finding Square Roots 507 pp. 507–510 Exs. 23, 26, 29, and every 3rd Ex. through Ex. 77, 79–81, 87, 95, 96, 104, 107, 110, 111 (with Ch. 8 Assess.) EXERCISE LEVELS Level A: Easier 23–38, 101–104 Level B: More Difficult 39–77, 79–88, 95, 96, 105–112 Level C: Most Difficult 78, 89–94, 97–100 HOMEWORK CHECK To quickly check student understanding of key concepts, go over the following exercises: Exs. 26, 38, 44, 50, 62, 74, 79. See also the Daily Homework Quiz: • Blackline Master (Chapter 9 Resource Book, p. 24) • Transparency (p. 68) COMMON ERROR EXERCISE 29 Some students may quickly write that the square roots of 0.09 are ±0.03. Remind them that when finding square roots of decimals less than one, the positive square root is larger than the original decimal. 507 EVALUATING EXPRESSIONS Use a calculator to evaluate the expression. Round the results to the nearest hundredth. STUDENT HELP COMMON ERROR EXERCISE 73 After subtracting 12 from each side, students will 5 likely multiply both sides by ᎏᎏ 4 before concluding that there is no solution. Point out that by being observant, they can save a step. After subtracting, the right side is negative and the coefficient of x is positive, so they know that the right side will remain negative without actually having to multiply. HOMEWORK HELP continued from p. 507 Example 4: Example 5: Example 6: Example 7: Exs. 45–53 Exs. 54–59 Exs. 60–77 Exs. 79–86 3 ± 4兹6苶 45. ᎏ 3 º2.27, 4.27 6 ± 4兹2苶 2 ± 5兹3苶 46. ᎏ º11.66, º0.34 47. ᎏ º1 5 º1.33, 2.13 1 ± 6兹8苶 48. ᎏ 6 º2.66, 3.00 7 ± 3兹2苶 º11.24, º2.76 50. 2 ± 5兹6苶 49. ᎏ ᎏ º1 2 º5.12, 7.12 5 ± 6兹3苶 51. ᎏ 3 º1.80, 5.13 3 ± 4兹5苶 52. ᎏ 4 º1.49, 2.99 7 ± 0.3兹1 苶2 苶 º1.34, º0.99 53. ᎏᎏ º6 QUADRATIC EQUATIONS Solve the equation or write no solution. Write the solutions as integers if possible. Otherwise write them as radical expressions. 54. x 2 = 36 º6, 6 55. b2 = 64 º8, 8 56. 5x 2 = 500 º10, 10 57. x 2 = 16 º4, 4 58. x 2 = 0 0 59. x 2 = º9 no solution 60. 3x 2 = 6 º兹2苶, 兹2苶 61. a 2 + 3 = 12 º3, 3 62. x 2 º 7 = 57 º8, 8 63. 2s 2 º 5 = 27 º4, 4 64. 5x 2 + 5 = 20 º兹3苶, 兹3苶 65. 7x 2 + 30 = 9 no solution 66. x 2 + 4.0 = 0 no solution 67. 6x 2 º 54 = 0 º3, 3 68. 7x 2 º 63 = 0 º3, 3 SOLVING EQUATIONS Use a calculator to solve the equation or write no solution. Round the results to the nearest hundredth. 69. 4x 2 º 3 = 57 70. 6y2 + 22 = 34 71. 2x 2 º 4 = 10 º3.87, 3.87 º1.41, 1.41 º2.65, 2.65 2 2 1 2 4 2 72. ᎏᎏn º 6 = 2 73. ᎏᎏx + 12 = 5 74. ᎏᎏx + 3 = 8 5 3 2 º3.46, 3.46 no solution º3.16, 3.16 2 2 75. 3x + 7 = 31 76. 6s º 12 = 0 77. 5a 2 + 10 = 20 º1.41, 1.41 º1.41, 1.41 º2.83, 2.83 78. CRITICAL THINKING Write your own example of a quadratic equation in the form x 2 = d for each of the following. 2 a. An equation that has no real solution. x = d, for any number d < 0 b. An equation that has one solution. x 2 = 0 80. Sample answer: t 0 1 1.5 1.75 1.8 1.9 1.95 c. An equation that has two solutions. x 2 = d, for any number d > 0 h 60 44 24 11 8.16 2.24 º0.84 about 1.95 sec ROAD SAFETY In Exercises 79–82, a boulder falls off the top of a cliff during a storm. The cliff is 60 feet high. Find how long it will take for the boulder to hit the road below. 79. Write the falling object model for s = 60. h = º16t 2 + 60 80. Try to determine the time when h = 0 from an input-output table for this model. See margin. 81. Solve the falling object model for h = 0. 兹3. 苶75 苶 ≈ 1.94 sec 82. Writing Which problem solving method do you prefer? Why? See margin. FALLING OBJECT MODEL In Exercises 83–86, an object is dropped from 82. Sample answer: I prefer solving the equation; it is a height s. How long does it take to reach the ground? Assume there is no more straightforward and air resistance. provides an exact 83. s = 144 feet 84. s = 256 feet 85. s = 400 feet 86. s = 600 feet answer. 3 sec 4 sec 5 sec about 6.1 sec 508 508 Chapter 9 Quadratic Equations and Functions INT STUDENT HELP NE ER T 87. HOMEWORK HELP Visit our Web site www.mcdougallittell.com for help with Exs. 87–88. 88. FOCUS ON CAREERS HOME COMPUTER SALES The sales S (in millions of dollars) of home computers in the United States from 1988 to 1995 can be modeled by S = 145.63t 2 + 3327.56, where t is the number of years since 1988. Use this model to estimate the year in which sales of home computers will be $36,000 million. 䉴 Source: Electronic Industries Association 2003 STUDENT HELP NOTES COMPUTER SOFTWARE SALES The sales S (in millions of dollars) of computer software in the United States from 1990 to 1995 can be modeled by S = 61.98t 2 + 1001.15, where t is the number of years since 1990. Use this model to estimate the year in which sales of computer software will be $7200 million. 䉴 Source: Electronic Industries Association 2000 MINERALS In Exercises 89–94, use the following information. Mineralogists use the Vickers scale to measure the hardness of minerals. The hardness H of a mineral can be determined by hitting the mineral with a pyramidshaped diamond and measuring the depth d of the indentation. The harder the mineral, the smaller the depth of the indentation. A model that relates mineral hardness with the indentation depth (in millimeters) is Hd 2 = 1.89. Use a calculator to find the depth of the indentation for the mineral with the given value of H. Round to the nearest hundredth of a millimeter. RE FE L AL I 90. Gold: H = 50 0.19 mm 91. Galena: H = 80 0.15 mm 92. Platinum: H = 125 0.12 mm 93. Copper: H = 140 0.12 mm 94. Hematite: H = 755 0.05 mm MINERALOGISTS The properties and distribution of minerals are studied by mineralogists. The Vickers scale applies to thin slices of minerals that can be examined with a microscope. INT 89. Graphite: H = 12 0.40 mm NE ER T CAREER LINK www.mcdougallittell.com Test Preparation 95. . APPLICATION NOTE EXERCISES 89–94 Although graphite has the lowest Vickers scale hardness number, 12, industrial diamonds may be synthesized by subjecting graphite to very high pressures and temperatures. The mineral chromite is at the high end of the scale, with a Vickers number of 1206. ENGLISH LEARNERS EXERCISES 87–94 Understanding the language of word problems can be extremely difficult for English language learners. In Exercises 87 and 88, you may need to clarify the usage of the word where, common in mathematical language but different from normal usage. For Exercises 89–94, make sure students understand both the vocabulary and the causeeffect relationship in the sentence The harder the mineral, the smaller the depth of the indentation. CAREER NOTE EXERCISES 89–94 Additional information about mineralogists is available at www.mcdougallittell.com. QUANTITATIVE COMPARISON In Exercises 95–96, choose the statement below that is true about the given numbers. A ¡ B ¡ C ¡ D ¡ Homework Help Students can find help for Exs. 87 and 88 at www.mcdougallittell.com. The information can be printed out for students who don’t have access to the Internet. The number in column A is greater. The number in column B is greater. The two numbers are equal. The relationship cannot be determined from the given information. For Lesson 9.1: Column A Column B º兹1苶2苶1苶 º2兹1苶2苶1苶 A The positive value of x in 2x 2 + 14 = 46 C 96. . The positive value of x in 2x 2 º 14 = 18 ADDITIONAL PRACTICE AND RETEACHING 9.1 Solving Quadratic Equations by Finding Square Roots 509 • Practice Levels A, B, and C (Chapter 9 Resource Book, p. 14) • Reteaching with Practice (Chapter 9 Resource Book, p. 17) • See Lesson 9.1 of the Personal Student Tutor For more Mixed Review: • Search the Test and Practice Generator for key words or specific lessons. 509 4 ASSESS DAILY HOMEWORK QUIZ Transparency Available 1. Find all square roots of 0.25. ±0.5 2. Evaluate –兹1. 苶44 苶. –1.2 2 3. Evaluate 兹b苶苶–苶苶4a苶c苶 for a = 3, b = 3, and c = –6. 9 4. Using a calculator, evaluate –3 ± 2兹5苶 ᎏ to the nearest 4 hundredth. –1.87, 0.37 SCIENCE ±1.53 7. A cliff diver jumps from an 84 ft cliff. Write a falling object model for the time t it will take the diver to hit the water. Assume there is no air resistance. h = –16t 2 + 84 1 2 that is, the exponential term in both equations is the same. 97. The NASA Lewis Research Center has two microgravity facilities. One provides a 132-meter drop into a hole and the other provides a 24-meter drop inside a tower. How long will each free-fall period be? about 5.19 sec, about 2.21 sec 98. In Japan a 490-meter-deep mine shaft has been converted into a microgravity facility. This creates the longest period of free fall currently available on Earth. How long will a period of free-fall be? 10 sec 99. If you want to double the free-fall time, how much do you have to increase the height from which the object was dropped? See margin. 100. CRITICAL THINKING How are these formulas similar? 1 2 h = º16t 2 + s when h is height, s is initial height, and t is time www.mcdougallittell.com MIXED REVIEW FACTORING INTEGERS Write the prime factorization. (Skills Review, p. 777) 102. 24 23 ª 3 103. 72 2 3 ª 32 104. 108 22 ª 33 GRAPH AND CHECK Use a graph to solve the linear system. Check your solution algebraically. (Review 7.1) EXTRA CHALLENGE NOTE Challenge problems for Lesson 9.1 are available in blackline format in the Chapter 9 Resource Book, p. 21 and at www.mcdougallittell.com. 105. º3x + 4y = º5 4x + 2y = º8 (º1, º2) 106. 4x + 5y = 20 5 ᎏᎏx + y = 4 4 (0, 4) 1 107. ᎏᎏx + 3y = 18 2 2x + 6y = º12 (º48, 14) SOLVING SYSTEMS In Exercises 109–111, use linear combinations to solve the system. (Review 7.3) ADDITIONAL TEST PREPARATION 1. WRITING Describe how to solve the quadratic equation ax 2 + c = 0. Sample answer: 108. 12x º 4y = º32 Add –c to each side. Then divide c each side by a to get x 2 = – }}. 冪莦莦 c c the solution ± – }} . If – }} is a a negative, there is no real solution. 510 º7x + y = 9 (º2, 2) (º4, º19) 110. 8x º 5y = 100 1 2x + ᎏᎏy = 4 2 (5, º12) BASKETBALL TICKETS You are selling tickets at a high school basketball game. Student tickets cost $2 and general admission tickets cost $3. You sell 2342 tickets and collect $5801. How many of each type of ticket did you sell? (Review 7.2) 1225 student tickets, 1117 general admission tickets 112. FLOWERS You are buying a combination of irises and white tulips for a flower arrangement. The irises are $1 each and the white tulips are $.50. You spend $20 total to purchase an arrangement of 25 flowers. How many of each kind did you purchase? (Review 7.2) 15 irises and 10 white tulips a square root of both sides to get 109. 10x º 3y = 17 x + 3y = 4 111. If – }} is nonnegative, take the 510 See margin. d = ᎏᎏg(t 2) when d is distance, g is gravity, and t is time EXTRA CHALLENGE 101. 11 11 c a In Exercises 97–100, use the following information. the equation d = ᎏᎏg(t 2), where g is the acceleration due to gravity 2 6. Solve 3y + 4 = 11. Write the result to the nearest hundredth. CONNECTION Scientists simulate a gravity-free environment called microgravity in free-fall situations. A similar microgravity environment can be felt on free-fall rides at amusement parks or when stepping off a high diving platform. The distance d (in meters) that an object that is dropped falls in t seconds can be modeled by (9.8 meters per second per second). 100. Sample answer: The first equation represents the distance a falling object falls in a given time t, and the second represents the height of the same object at time t. Then h = s º d and 1 so º16t 2 = º}}g(t 2); 2 5. Solve x 2 – 7 = 13. Write the result as a radical expression. ±兹20 苶 ★ Challenge 99. 4 times; the distance that an object is dropped is a function of the time squared. So (2t)2 = 22 • t 2 = 4t 2 Chapter 9 Quadratic Equations and Functions