9-6 The Quadratic Formula and the Discriminant Content Standards A.REI..a Use the method of completing the square to transform an quadratic equation in into an equation of the form ( p) 5 q... Derive the quadratic formula from this form. Also N.Q.3, A.CED.1, A.REI..b Objectives To solve quadratic equations using the quadratic formula To find the number of solutions of a quadratic equation You have several was to solve this problem: graphing, making a table, and factoring. Which will ou use? Your friend s aunt has a brick walkwa in her backard. Her plan is to decrease the length b the same amount she increases the width to make a rectangular patio. She wants the patio to have an area of 310 ft. Can she build a patio to meet her plan? Eplain our reasoning. 5 ft 30 ft Lesson Vocabular quadratic formula discriminant MATHEMATICAL PRACTICES Recall that quadratic equations can have two, one, or no real-number solutions. A quadratic equation can never have more than two solutions. Essential Understanding You can find the solution(s) of an quadratic equation using the quadratic formula. Ke Concept Quadratic Formula Algebra If a 1 b 1 c 5 0, and a 0, then 5 b "b ac a Eample Suppose 1 3 5 5 0. Then a 5, b 5 3, and c 5 5. Therefore 5 (3) "(3) ()(5) () 58 Chapter 9 Quadratic Functions and Equations
Here s Wh It Works If ou complete the square for the general equation a 1 b 1 c 5 0, ou can derive the quadratic formula. Step 1 Write a 1 b 1 c 5 0 so the coefficient of is 1. Step a 1 b 1 c 5 0 1 a b 1 a c 5 0 Divide each side b a. Complete the square. 1 a b 5 c a 1 b a 1 Q b a R c 5 a 1 Q b a R Q 1 b a R c 5 a 1 b a Q 1 b a R ac 5 a 1 b a Subtract a c from each side. b Add Q a R to each side. Write the left side as a square. Multipl c a Q 1 b a R b ac 5 a Simplif the right side. Step 3 Solve the equation for. a b a to get like denominators. This step uses the propert m n m n which ou will stud in Lesson 10-., Å Q 1 b a R 5 Å b ac a Take square roots of each side. 1 b a 5 "b ac a 5 b a "b ac a 5 b "b ac a Simplif the right side. Subtract b a from each side. Simplif. Be sure to write a quadratic equation in standard form before using the quadratic formula. Wh do ou need to write the equation in standard form? You can onl use the quadratic formula with equations in the form a 1 b 1 c 5 0. Problem 1 Using the Quadratic Formula What are the solutions of 8 5? Use the quadratic formula. 8 5 0 5 b "b ac a 5 () "() (1)(8) (1) 5!36 Write the equation in standard form. Use the quadratic formula. Substitute 1 for a, for b, and 8 for c. Simplif. 5 1 6 or 5 6 Write as two equations. 5 or 5 Simplif. Got It? 1. What are the solutions of 5 1? Use the quadratic formula. Lesson 9-6 The Quadratic Formula and the Discriminant 583
When the radicand in the quadratic formula is not a perfect square, ou can use a calculator to approimate the solutions of an equation. Problem Finding Approimate Solutions Sports In the shot put, an athlete throws a heav metal ball through the air. The arc of the ball can be modeled b the equation 5 0.0 1 0.8 1, where is the horizontal distance, in meters, from the athlete and is the height, in meters, of the ball. How far from the athlete will the ball land? 6 Wh do ou substitute 0 for? When the ball hits the ground, its height will be 0. 0 0 5 10 15 0 0 5 0.0 1 0.8 1 Substitute 0 for in the given equation. 5 b "b ac a 5 0.8 "0.8 (0.0)() (0.0) 5 0.8 "1.056 0.08 Use the quadratic formula. Substitute 0.0 for a, 0.8 for b, and for c. Simplif. 0.8 1 "1.056 0.8 "1.056 5 0.08 or 5 0.08 Write as two equations. <.16 or < 3.16 Simplif. Onl the positive answer makes sense in this situation. The ball will land about 3.16 m from the athlete. Got It?. A batter strikes a baseball. The equation 5 0.005 1 0.7 1 3.5 models its path, where is the horizontal distance, in feet, the ball travels and is the height, in feet, of the ball. How far from the batter will the ball land? Round to the nearest tenth of a foot. There are man methods for solving a quadratic equation. Method Graphing Square roots Factoring Completing the square Quadratic formula When to Use Use if ou have a graphing calculator hand. Use if the equation has no -term. Use if ou can factor the equation easil. Use if the coefficient of is 1, but ou cannot easil factor the equation. Use if the equation cannot be factored easil or at all. 58 Chapter 9 Quadratic Functions and Equations
Problem 3 Choosing an Appropriate Method Can ou use the quadratic formula to solve part (A)? Yes. You can use the quadratic formula with a 5 3, b 5 0, and c 5 9. However, it is faster to use square roots. Which method(s) would ou choose to solve each equation? Eplain our reasoning. A 3 9 5 0 Square roots; there is no -term B Factoring; the equation is easil factorable 30 5 0 C 6 1 13 17 5 0 Quadratic formula, graphing; the equation cannot be factored D Quadratic formula, completing the square, or graphing; the coefficient of the -term is 1, but the equation cannot be factored 5 1 3 5 0 E 16 50 1 1 5 0 Quadratic formula, graphing; the equation cannot be factored easil since the numbers are large Got It? 3. Which method(s) would ou choose to solve each equation? Justif our reasoning. a. 8 1 1 5 0 b. 169 5 36 c. 5 1 13 1 5 0 Quadratic equations can have two, one, or no real-number solutions. Before ou solve a quadratic equation, ou can determine how man real-number solutions it has b using the discriminant. The discriminant is the epression under the radical sign in the quadratic formula. 5 b "b ac a the discriminant The discriminant of a quadratic equation can be positive, zero, or negative. Ke Concept b ac. 0 b ac 5 0 6 1 7 5 0 The discriminant is (6) (1)(7) 5 8, which is positive. 6 1 9 5 0 The discriminant is (6) (1)(9) 5 0. Discriminant Eample Using the Discriminant O 3 6 1 11 5 0 The discriminant is (6) (1)(11) 5 8, which is negative. O Number of Solutions b ac, 0 O There are two realthere is one realthere are no realnumber solutions. number solution. number solutions. hsm11a1se_0906_t05551 hsm11a1se_0906_t0555.ai Lesson 9-6 The Quadratic Formula and the Discriminant 058_hsm1a1se_0906.indd 585 585 3/3/11 8:5:7 AM
Problem Using the Discriminant How man real-number solutions does 3 5 5 have? Can ou solve this problem another wa? Yes. You could actuall solve the equation to find an solutions. However, ou onl need to know the number of solutions, so use the discriminant. Write the equation in standard form. 3 1 5 5 0 Evaluate the discriminant b substituting for a, 3 for b, and 5 for c. b ac 5 (3) ()(5) 5 31 Because the discriminant is negative, the equation has no real-number solutions. Draw a conclusion. Got It?. a. How man real-number solutions does 6 5 5 7 have? b. Reasoning If a is positive and c is negative, how man real-number solutions will the equation a 1 b 1 c 5 0 have? Eplain. Lesson Check Do ou know HOW? Do ou UNDERSTAND? MATHEMATICAL PRACTICES. Vocabular Eplain how the discriminant of the equation a 1 b 1 c 5 0 is related to the number of -intercepts of the graph of 5 a 1 b 1 c. Use the quadratic formula to solve each equation. If necessar, round answers to the nearest hundredth. 1. 3 11 1 5 0. 7 5 8 5. Reasoning What method would ou use to solve the equation 1 9 1 c 5 0 if c 5 1? If c 5 7? Eplain. 3. How man real-number solutions does the equation 1 8 5 5 0 have? 6. Writing Eplain how completing the square is used to derive the quadratic formula. Practice and Problem-Solving Eercises A Practice 586 Chapter 9 058_hsm1a1se_0906.indd 586 MATHEMATICAL PRACTICES Use the quadratic formula to solve each equation. See Problem 1. 7. 1 5 1 3 5 0 8. 5 1 16 8 5 0 9. 1 7 15 5 0 10. 3 1 5 110 11. 18 5 50 5 0 1. 3 1 5 96 13. 3 1 19 5 15 1. 10 5 0 15. 5 7 5 156 Quadratic Functions and Equations 3/3/11 8:5:9 AM
Use the quadratic formula to solve each equation. Round our answer to the nearest hundredth. See Problem. 16. 1 8 1 11 5 0 17. 5 1 1 5 0 18. 16 5 5 19. 8 7 5 5 0 0. 6 1 9 5 3 1. 3 1 5 5. Football A football plaer punts a ball. The path of the ball can be modeled b the equation 5 0.00 1 1.5, where is the horizontal distance, in feet, the ball travels and is the height, in feet, of the ball. How far from the football plaer will the ball land? Round to the nearest tenth of a foot. Which method(s) would ou choose to solve each equation? Justif our reasoning. See Problem 3. B Appl 3. 1 15 5 0. 9 9 5 0 5. 1 5 73 6. 3 7 1 3 5 0 7. 1 60 5 0 8. 1 8 1 1 5 0 Find the number of real-number solutions of each equation. 9. 1 3 5 0 30. 1 7 5 5 0 31. 1 3 1 11 5 0 3. 15 5 0 33. 1 5 0 3. 9 1 1 1 5 0 Use an method to solve each equation. If necessar, round our answer to the nearest hundredth. 35. 3w 5 8 36. 3 1 5 0 37. 6g 18 5 0 38. 3p 1 p 5 10 39. k k 5 0. 13r 117 5 0 1. Think About a Plan You operate a dog-walking service. You have 50 customers per week when ou charge $1 per walk. For each $1 decrease in our fee for walking a dog, ou get 5 more customers per week. Can ou ever earn $750 in a week? Eplain. What quadratic equation in standard form can ou use to model this situation? How can the discriminant of the equation help ou solve the problem?. Sports Your school wants to take out an ad in the paper congratulating the basketball team on a successful season, as shown below. The area of the photo will be half the area of the entire ad. What is the value of? See Problem. 7 in. Photo 5 in. 3. Writing How can ou use the discriminant to write a quadratic equation that has two solutions?. Error Analsis Describe and correct the error at the right that a student made in finding the discriminant of 1 5 6 5 0. a =, b = 5, c = 6 b ac = 5 ()(-6) = 5 8 = 3 Lesson 9-6 The Quadratic Formula and the Discriminant 587
C Challenge 5. Find the discriminant and the solution of each equation in parts (a) (c). If necessar, round to the nearest hundredth. a. 6 1 5 5 0 b. 1 0 5 0 c. 7 3 5 0 d. Reasoning When the discriminant is a perfect square, are the solutions rational or irrational? Eplain. 6. Reasoning The solutions of an quadratic equation a 1 b 1 c 5 0 are b 1 "b ac a and b "b ac a. a. Find a formula for the sum of the solutions. b. One solution of 1 3 10 5 0 is 8. Use the formula ou found in part (a) to find the second solution. Reasoning For each condition given, tell whether a 1 b 1 c 5 0 will alwas, sometimes, or never have two solutions. 7. b, ac 8. b 5 0 9. ac, 0 Standardized Test Prep SAT/ACT 50. What are the approimate solutions of the equation 7 1 3 5 0? 6.5, 0.6 6.5, 0.6 0.6, 6.5 0.6, 6.5 51. Which of the following relations is a function? 5(1, ), (3, 5), (1, ), (, 3)6 5(8, ), (6, 3), (6, 11), (8, )6 5(5, 6), (0, 9), (1, ), (0, 6)6 5(1, 3), (7, 3), (7, ), (, 5)6 5. What equation do ou get when ou solve 3a b 5 c for b? b 5 3a 1 c b 5 3a c b 5 3a 1 c b 5 3a c 53. What are the approimate solutions of the equation 1 3 5 1 1 5 0? Use a graphing calculator. 1.07,.77 1.16,.59 0.87, 10.38 0.19, 16.01 Short Response 5. Suppose the line through points (n, 6) and (1, ) is parallel to the graph of 1 5 3. Find the value of n. Show our work. Mied Review Solve each equation b completing the square. 55. s 10s 1 13 5 0 56. m 1 3m 5 57. 3w 1 18w 1 5 0 Get Read! To prepare for Lesson 9-7, do Eercises 58 61. Graph each function. 58. 5 59. 5 3 1 60. 5 Q 3 R 1 61. 5 Q R See Lesson 9-5. See Lesson 7-6. 588 Chapter 9 Quadratic Functions and Equations