Solve Quadratic Equations by the Quadratic Formula. The solutions of the quadratic equation ax 2 1 bx 1 c 5 0 are. Standardized Test Practice

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1 10.6 Solve Quadratic Equations by the Quadratic Formula Before You solved quadratic equations by completing the square. Now You will solve quadratic equations using the quadratic formula. Why? So you can solve a problem about film production, as in Example 3. Key Vocabulary quadratic formula By completing the square for the quadratic equation ax 2 1 bx 1 c 5 0, you can develop a formula that gives the solutions of any quadratic equation in standard form. This formula is called the quadratic formula. (The quadratic formula is developed on page 727.) KEY CONCEPT For Your Notebook The Quadratic Formula The solutions of the quadratic equation ax 2 1 bx 1 c 5 0 are x 5 2b 6 Î } b 2 2 4ac }} where a Þ 0 and b 2 2 4ac 0. E XAMPLE 1 Standardized Test Practice What are the solutions of 3x 2 1 5x 5 8? A 21 and 2 8 } 3 B 21 and 8 } 3 C 1and2 8 } 3 D 1 and 8 } 3 ANOTHER WAY Instead of solving the equation, you can check the answer choices in the equation. Solution 3x 2 1 5x 5 8 3x 2 1 5x x5 2b 6 Ï} b 2 2 4ac }} Write original equation. Write in standard form. Quadratic formula 25 6 Ï}} 5 x (3)(28) Substitute values in the quadratic formula: 2(3) a 5 3, b 5 5, and c Ï} 121 Simplify Simplify the square root. The solutions of the equation are c The correct answer is C. A B C D and } Solve Quadratic Equations by the Quadratic Formula 671

2 E XAMPLE 2 Solve a quadratic equation Solve 2x x. 2x x 2x 2 2 x Write original equation. Write in standard form. x 5 2b 6 Ï} b 2 2 4ac }} Quadratic formula 2(21) 6 Ï}} (21) (2)(27) Substitute values in the quadratic }}} 2(2) formula: a 5 2, b 521, and c Ï} 57 } 4 Simplify. c The solutions are 1 1 Ï} 57 } 4 ø 2.14 and 1 2 Ï} 57 } 4 ø at classzone.com CHECK Write the equation in standard form, 2x 2 2 x Then graph the related function y 5 2x 2 2 x 2 7. The x-intercepts are about 21.6 and 2.1. So, each solution checks GUIDED PRACTICE for Examples 1 and 2 Use the quadratic formula to solve the equation. Round your solutions to the nearest hundredth, if necessary. 1. x 2 2 8x n 2 2 5n z 2 5 7z 1 2 E XAMPLE 3 Use the quadratic formula FILM PRODUCTION For the period , the number y of films produced in the world can be modeled by the function y 5 10x x where x is the number of years since In what year were 4200 films produced? Solution y 5 10x x Write function x x Substitute 4200 for y x x Write in standard form. INTERPRET SOLUTIONS The solution 23 can be ignored because 23 represents the year 1968, which is not in the given time period. 2(294) 6 Ï}} (294) x (10)(2300) Substitute values in the quadratic }}} 2(10) formula: a 5 10, b 5294, and c Ï } 20,836 }} 20 Simplify. The solutions of the equation are 94 1 Ï } 20,836 }} ø 12 and 94 2 Ï } 20,836 }} ø c There were 4200 films produced about 12 years after 1971, or in Chapter 10 Quadratic Equations and Functions

3 GUIDED PRACTICE for Example 3 4. WHAT IF? In Example 3, find the year when 4750 films were produced. CONCEPT SUMMARY For Your Notebook Methods for Solving Quadratic Equations Method Lesson(s) When to Use Factoring Use when a quadratic equation can be factored easily. Graphing 10.3 Use when approximate solutions are adequate. Finding square roots 10.4 Use when solving an equation that can be written in the form x 2 5 d. Completing the square 10.5 Can be used for any quadratic equation ax 2 1 bx 1 c 5 0 but is simplest to apply when a 5 1 and b is an even number. Quadratic formula 10.6 Can be used for any quadratic equation. E XAMPLE 4 Choose a solution method Tell what method you would use to solve the quadratic equation. Explain your choice(s). a. 10x b. x 2 1 4x 5 0 c. 5x 2 1 9x Solution a. The quadratic equation can be solved using square roots because the equation can be written in the form x 2 5 d. b. The equation can be solved by factoring because the expression x 2 1 4x can be factored easily. Also, the equation can be solved by completing the square because the equation is of the form ax 2 1 bx 1 c 5 0 where a 5 1 and b is an even number. c. The quadratic equation cannot be factored easily, and completing the square will result in many fractions. So, the equation can be solved using the quadratic formula. GUIDED PRACTICE for Example 4 Tell what method you would use to solve the quadratic equation. Explain your choice(s). 5. x 2 1 x x x 2 1 6x Solve Quadratic Equations by the Quadratic Formula 673

4 10.6 EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS1 for Exs. 19 and 47 5 STANDARDIZED TEST PRACTICE Exs. 2, 12, 25, and 50 5 MULTIPLE REPRESENTATIONS Ex VOCABULARY What formula can be used to solve any quadratic equation? 2. WRITING What method(s) would you use to solve 2x 2 1 8x 5 1? Explain your choice(s). EXAMPLES 1 and 2 on pp for Exs SOLVING QUADRATIC EQUATIONS Use the quadratic formula to solve the equation. Round your solutions to the nearest hundredth, if necessary. 3. x 2 1 5x x 2 2 x x 2 2 2x m 2 1 3m z 2 1 z n 2 2 5n w w t 2 1 3t g 2 1 9g MULTIPLE CHOICE What are the solutions of 10x 2 2 3x ? A 2} 1 and 2} 1 B 2} 1 and } 1 C } 1 and 2} 1 D } 1 and } SOLVING QUADRATIC EQUATIONS Use the quadratic formula to solve the equation. Round your solutions to the nearest hundredth, if necessary. 13. x 2 2 5x x x x 2 2 2x 16. 2m 2 1 9m r r g 2 2 6g g 19. 6z 2 5 2z 2 1 7z h h 21. 4t 2 2 3t t y 2 2 3y y n n w w MULTIPLE CHOICE What are the solutions of x x 5 2x 2 11? A 22 and 222 B 21 and 211 C 1 and 11 D 2 and 22 ERROR ANALYSIS Describe and correct the error in solving the equation x 2 2 5x x 2 1 3x 5 1 x Ï }} (25) 2 2 4(7)(21) }} 2(7) Ï} 53 } 14 x ø and x ø 0.16 x Ï }} (22)(1) }} 2(22) Ï} 17 } 24 x ø and x ø 1.78 EXAMPLE 4 on p. 673 for Exs CHOOSING A METHOD Tell what method(s) you would use to solve the quadratic equation. Explain your choice(s) x x x x m 2 1 5m z 2 2 4z g g Chapter 10 Quadratic Equations and Functions

5 SOLVING QUADRATIC EQUATIONS Solve the quadratic equation using any method. Round your solutions to the nearest hundredth, if necessary x x 2 2 8x x 2 1 2x x x x 2 1 4x x 2 1 x x x x x x 42. (x 1 13) GEOMETRY Use the given area A of the rectangle to find the value of x. Then give the dimensions of the rectangle. 43. A 5 91 m A ft 2 (x 1 2) m (4x 2 5) ft (2x 1 3) m (4x 1 3) ft 45. CHALLENGE The solutions of the quadratic equation ax 2 1 bx 1 c 5 0 are x 5 2b 1Ï} b 2 2 4ac }} and x 5 2b 2Ï} b 2 2 4ac }}. Find the mean of the solutions. How is the mean of the solutions related to the graph of y 5 ax 2 1 bx 1 c? Explain. PROBLEM SOLVING EXAMPLE 3 on p. 672 for Exs ADVERTISING For the period , the amount of money y (in billions of dollars) spent on advertising in the U.S. can be modeled by the function y x x where x is the number of years since In what year was 164 billion dollars spent on advertising? 47. CELL PHONES For the period , the number y (in millions) of cell phone service subscribers in the U.S. can be modeled by the function y 5 0.7x x where x is the number of years since In what year were there 16,000,000 cell phone service subscribers? 48. MULTI-STEP PROBLEM A football is punted from a height of 2.5 feet above the ground and with an initial vertical velocity of 45 feet per second. Not drawn to scale 2.5 ft 5.5 ft a. Use the vertical motion model to write an equation that gives the height h (in feet) of the football as a function of the time t (in seconds) after it has been punted. b. The football is caught 5.5 feet above the ground as shown in the diagram. Find the amount of time that the football is in the air Solve Quadratic Equations by the Quadratic Formula 675

6 49. MULTIPLE REPRESENTATIONS For the period , the number y (in thousands) of 16- and 17-year-olds employed in the United States can be modeled by the function y x x where x is the number of years since a. Solving an Equation Write and solve an equation to find the year during which 2,500, and 17-year-olds were employed. b. Drawing a Graph Graph the function on a graphing calculator. Use the trace feature to find the year when 2,500, and 17-year-olds were employed. Use the graph to check your answer from part (a). 50. SHORT RESPONSE NASA creates a weightless environment by flying a plane in a series of parabolic paths. The height h (in feet) of a plane after t seconds in a parabolic flight path can be modeled by the graph of h 5211t t 1 21,000. The passengers experience a weightless environment when the height of the plane is greater than or equal to 30,800 feet. Find the period of weightlessness on such a flight. Explain. 51. CHALLENGE Mineral deposits have formed a uniform coating that is 4 millimeters thick on the inside of a water pipe. The cross-sectional area of the pipe has decreased by 10%. What was the original diameter of the pipe (to the nearest tenth of a millimeter)? MIXED REVIEW PREVIEW Prepare for Lesson 10.7 in Exs Evaluate the expression x 2 when x 5 2 (p. 8) w } 2w when w 5 10 (p. 103) Ï } x when x (p. 110) Ï } x when x 5 49 (p. 110) Graph the equation. 56. x 5 8 (p. 215) 57. 3x 2 y 5 2 (p. 225) 58. y 52 2 } 5 x 2 6 (p. 244) 59. y 527x 2 (p. 628) 60. y 5 8x (p. 628) 61. y 52x 2 2 6x 1 5 (p. 635) QUIZ for Lessons Solve the equation using square roots. (p. 652) 1. 3x x x Solve the equation by completing the square. (p. 663) 4. x 2 1 2x x x x 2 2 8x x x x 2 2 5x 52} x 2 1 x Solve the equation using the quadratic formula. (p. 671) 10. x 2 1 4x x 2 1 3x x x EXTRA PRACTICE for Lesson 10.6, p. 947 ONLINE QUIZ at classzone.com

7 ACTIVITY Investigating Algebra Use before Lesson The Discriminant QUESTION How can you determine the number of solutions of a quadratic equation? In the quadratic formula, x 5 2b 6 Ï} b 2 2 4ac }}, the expression b 2 2 4ac is called the discriminant. E XPLORE Determine how the discriminant is related to the number of solutions of a quadratic equation STEP 1 Find the number of solutions Find the number of solutions of the equations below by finding the number of x-intercepts of the graphs of the related functions. 0 5 x 2 2 6x x 2 2 6x x 2 2 6x 1 12 STEP 2 Find the value of b 2 2 4ac For each equation in Step 1, determine whether the value of b 2 2 4ac is positive, negative, or zero. 2 y y 5 x 2 2 6x 1 12 y 5 x 2 2 6x x STEP 3 Make a table Organize your results from Steps 1 and 2 in a table as shown. Equation Number of solutions Value of b 2 2 4ac y 5 x 2 2 6x x 2 2 6x 2 7?? 0 5 x 2 2 6x 1 9?? 0 5 x 2 2 6x 1 12?? STEP 4 Make a conjecture Make a generalization about the value of the discriminant and the number of solutions of a quadratic equation. DRAW CONCLUSIONS Use your observations to complete these exercises 1. Repeat Steps 123 using the following equations: x 2 1 4x , x 2 1 4x , and x 2 1 4x Is your conjecture still true? 2. Notice that the expression b 2 2 4ac is under the radical sign in the quadratic formula. Use this observation to explain why the value of b 2 2 4ac determines the number of solutions of a quadratic equation Interpret the Discriminant 677