10.7. Interpret the Discriminant. For Your Notebook. x5 2b 6 Ï} b 2 2 4ac E XAMPLE 1. Use the discriminant KEY CONCEPT

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1 10.7 Interpret the Discriminant Before You used the quadratic formula. Now You will use the value of the discriminant. Wh? So ou can solve a problem about gmnastics, as in E. 49. Ke Vocabular discriminant In the quadratic formula, the epression b 2 2 4ac is called the discriminant of the associated equation a 2 1 b 1 c b 6 Ï} b 2 2 4ac }} 2a discriminant Because the discriminant is under the radical smbol, the value of the discriminant can be used to determine the number of solutions of a quadratic equation and the number of -intercepts of the graph of the related function. KEY CONCEPT For Your Notebook Using the Discriminant of a 2 1 b 1 c 5 0 READING Recall that in this course, solutions refers to real-number solutions. Value of the discriminant Number of solutions Graph of 5 a 2 1 b 1 c b 2 2 4ac > 0 b 2 2 4ac 5 0 b 2 2 4ac < 0 Two solutions One solution No solution Two -intercepts One -intercept No -intercept E XAMPLE 1 Use the discriminant Equation Discriminant Number of a 2 + b + c = 0 b 2 4ac solutions a (2)(5) 524 No solution b (1)(27) 5 28 Two solutions c (212) 2 2 4(4)(9) 5 0 One solution 678 Chapter 10 Quadratic Equations and Functions

2 E XAMPLE 2 Find the number of solutions Tell whether the equation has two solutions, one solution, or no solution. Solution STEP 1 Write the equation in standard form Write equation Subtract 2 from each side. STEP 2 Find the value of the discriminant. b 2 2 4ac 5 (22) 2 2 4(3)(27) Substitute 3 for a, 22 for b, and 27 for c Simplif. c The discriminant is positive, so the equation has two solutions. GUIDED PRACTICE for Eamples 1 and 2 Tell whether the equation has two solutions, one solution, or no solution E XAMPLE 3 Find the number of -intercepts Find the number of -intercepts of the graph of Solution Find the number of solutions of the equation b 2 2 4ac 5 (5) 2 2 4(1)(8) Substitute 1 for a, 5 for b, and 8 for c. 527 Simplif. c The discriminant is negative, so the equation has no solution. This means that the graph of has no -intercepts. CHECK You can use a graphing calculator to check the answer. Notice that the graph of has no -intercepts. GUIDED PRACTICE for Eample 3 Find the number of -intercepts of the graph of the function Interpret the Discriminant 679

3 E XAMPLE 4 Solve a multi-step problem FOUNTAINS The Centennial Fountain in Chicago shoots a water arc that can be modeled b the graph of the equation where is the horizontal distance (in feet) from the river s north shore and is the height (in feet) above the river. Does the water arc reach a height of 50 feet? If so, about how far from the north shore is the water arc 50 feet above the water? 50 North shore 50 Solution STEP 1 Write a quadratic equation. You want to know whether the water arc reaches a height of 50 feet, so let Then write the quadratic equation in standard form Write given equation Substitute 50 for Subtract 50 from each side. STEP 2 Find the value of the discriminant of b 2 2 4ac 5 (1.2) 2 2 4(20.006)(240) a , b 5 1.2, c Simplif. STEP 3 Interpret the discriminant. Because the discriminant is positive, the equation has two solutions. So, the water arc reaches a height of 50 feet at two points on the water arc. STEP 4 Solve the equation to find the distance from the north shore where the water arc is 50 feet above the water. USE A SHORTCUT Because the value of b 2 4ac was calculated in Step 2, ou can substitute 0.48 for b 2 4ac. 5 2b 6 Ï} b 2 2 4ac }} 2a Quadratic formula Ï} 0.48 } 2(20.006) Substitute values in the quadratic formula. ø 42 or ø 158 Use a calculator. c The water arc is 50 feet above the water about 42 feet from the north shore and about 158 feet from the north shore. GUIDED PRACTICE for Eample 4 7. WHAT IF? In Eample 4, does the water arc reach a height of 70 feet? If so, about how far from the north shore is the water arc 70 feet above the water? 680 Chapter 10 Quadratic Equations and Functions

4 10.7 EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS1 for Es. 9 and 47 5 STANDARDIZED TEST PRACTICE Es. 2, 18, 19, 40, 41, and VOCABULARY Write the quadratic formula and circle the epression that represents the discriminant. 2. WRITING Eplain how the discriminant of a 2 1 b 1 c 5 0 is related to the graph of 5 a 2 1 b 1 c. EXAMPLES 1 and 2 on pp for Es USING THE DISCRIMINANT Tell whether the equation has two solutions, one solution, or no solution m 2 2 6m v 2 2 6v q p p h h } 4 z z g 2 2 4g 5 4 } r r r 15. 3n n 2 3n w 2 2 7w w 18. MULTIPLE CHOICE What is the value of the discriminant of the equation ? A 29 B 9 C 59 D MULTIPLE CHOICE How man solutions does have? A None B One C Two D Three ERROR ANALYSIS Describe and correct the error in finding the number of solutions of the equation b 2 2 4ac (4)(9) The equation has two solutions. b 2 2 4ac 5 (27) 2 2 4(3)(24) (248) 5 97 The equation has two solutions. EXAMPLE 3 on p. 679 for Es FINDING THE NUMBER OF -INTERCEPTS Find the number of -intercepts of the graph of the function } } REASONING Give a value of c for which the equation has (a) two solutions, (b) one solution, and (c) no solution c c c Interpret the Discriminant 681

5 USING THE DISCRIMINANT Tell whether the verte of the graph of the function lies above, below, or on the -ais. Eplain our reasoning OPEN ENDED Write a function of the form 5 a 2 1 b 1 c whose graph has one -intercept. 41. EXTENDED RESPONSE Use the rectangular prism shown. a. The surface area of the prism is 314 square meters. Write an equation that ou can solve to find the value of w. b. Use the discriminant to determine the number of values of w in the equation from part (a). c. Solve the equation. Do the value(s) of w make sense in the contet of the problem? Eplain. (w 1 4) m 8 m w m CHALLENGE Find all values of k for which the equation has (a) two solutions, (b) one solution, and (c) no solution k k k PROBLEM SOLVING EXAMPLE 4 on p. 680 for Es BIOLOGY The amount (in milliliters per gram of bod mass per hour) of ogen consumed b a parakeet during flight can be modeled b the function where is the speed (in kilometers per hour) of the parakeet. a. Use the discriminant to show that it is possible for a parakeet to consume 25 milliliters of ogen per gram of bod mass per hour. b. Find the speed(s) at which the parakeet consumes 25 milliliters of ogen per gram of bod mass per hour. Round our solution(s) to the nearest tenth. 46. FOOD For the period , the average amount (in pounds per person per ear) of butter consumed in the United States can be modeled b where is the number of ears since According to the model, did the butter consumption in the United States ever reach 5 pounds per person per ear? If so, in what ear(s)? 47. SHORT RESPONSE The frame of the tent shown is defined b a rectangular base and two parabolic arches that connect the opposite corners of the base. The graph of models the height (in feet) of one of the arches feet along the diagonal of the base. Can a child that is 4 feet tall walk under one of the arches without having to bend over? Eplain WORKED-OUT SOLUTIONS on p. WS1 5 STANDARDIZED TEST PRACTICE

6 48. SCIENCE Between the months of April and September, the number of hours of dalight per da in Seattle, Washington, can be modeled b where is the number of das since April 1. a. Do an of the das between April and September in Seattle have 17 hours of dalight? If so, how man? b. Do an of the das between April and September in Seattle have 14 hours of dalight? If so, how man? 49. MULTI-STEP PROBLEM During a trampoline competition, a trampolinist leaves the mat when her center of gravit is 6 feet above the ground. She has an initial vertical velocit of 32 feet per second. a. Use the vertical motion model to write an equation that models the height h (in feet) of the center of gravit of the trampolinist as a function of the time t (in seconds) into her jump. b. Does her center of gravit reach a height of 24 feet during the jump? If so, at what time(s)? c. On another jump, the trampolinist leaves the mat when her center of gravit is 6 feet above the ground and with an initial vertical velocit of 35 feet per second. Does her center of gravit reach a height of 24 feet on this jump? If so, at what time(s)? 50. CHALLENGE Last ear, a manufacturer sold backpacks for $24 each. At this price, the manufacturer sold about 1000 backpacks per week. A marketing analst predicts that for ever $1 reduction in the price of the backpack, the manufacturer will sell 100 more backpacks per week. a. Write a function that models the weekl revenue R (in dollars) that the manufacturer will receive for reductions of $1 in the price of the backpack. b. Is it possible for the manufacturer to receive a weekl revenue of $28,000? $30,000? What is the maimum weekl revenue that the manufacturer can receive? Eplain our answers using the discriminants of quadratic equations. h ft MIXED REVIEW PREVIEW Prepare for Lesson 10.8 in Es Graph the function (p. 225) } 4 (p. 244) } (p. 244) (p. 520) (0.2) (p. 531) (p. 628) Solve the equation. 57. a (p. 134) 58. f (p. 134) 59. 4z (p. 141) 60. 9w (p. 141) 61. 2b 2 b (p. 148) ( 2 4) 5 9 (p. 148) Solve the equation b factoring. (p. 593) n 2 1 2n a a EXTRA PRACTICE for Lesson 10.7, p. 947 ONLIN E QUIZ a t classzone.com 683