Essential Question How can you use completing the square to solve a quadratic equation?

Transcription

1 9.4 Solving Quadratic Equations Completing the Square Essential Question How can ou use completing the square to solve a quadratic equation? Work with a partner. a. Write the equation modeled the algera tiles. This is the equation to e solved. Solving Completing the Square. Four algera tiles are added to the left side to complete the square. Wh are four algera tiles also added to the right side? MAKING SENSE OF PROBLEMS To e proficient in math, ou need to eplain to ourself the meaning of a prolem. After that, ou need to look for entr points to its solution. c. Use algera tiles to lael the dimensions of the square on the left side and simplif on the right side. d. Write the equation modeled the algera tiles so that the left side is the square of a inomial. Solve the equation using square roots. Work with a partner. a. Write the equation modeled the algera tiles. Solving Completing the Square. Use algera tiles to complete the square. c. Write the solutions of the equation. d. Check each solution in the original equation. Communicate Your Answer 3. How can ou use completing the square to solve a quadratic equation? 4. Solve each quadratic equation completing the square. a c Section 9.4 Solving Quadratic Equations Completing the Square 505

2 9.4 Lesson What You Will Learn Core Vocaular completing the square, p. 506 Previous perfect square trinomial coefficient maimum value minimum value verte form of a quadratic function JUSTIFYING STEPS In each diagram elow, the comined area of the shaded regions is +. Adding ( ) completes the square in the second diagram. Complete the square for epressions of the form +. Solve quadratic equations completing the square. Find and use maimum and minimum values. Solve real-life prolems completing the square. Completing the Square For an epression of the form +, ou can add a constant c to the epression so that + + c is a perfect square trinomial. This process is called completing the square. Core Concept Completing the Square Words To complete the square for an epression of the form +, follow these steps. Step 1 Find one-half of, the coefficient of. Step Square the result from Step 1. Step 3 Add the result from Step to +. Factor the resulting epression as the square of a inomial. Algera + + ( ) ( + ) Completing the Square Complete the square for each epression. Then factor the trinomial. a a. Step 1 Find one-half of. 6 3 Step Square the result from Step Step 3 Add the result from Step to ( ) ( ) ( ) ( + 3). Step 1 Find one-half of. 9 Step Square the result from Step 1. ( 9 ) 81 4 Step 3 Add the result from Step to ) ( 9 Monitoring Progress Help in English and Spanish at BigIdeasMath.com Complete the square for the epression. Then factor the trinomial Chapter 9 Solving Quadratic Equations

3 Solving Quadratic Equations Completing the Square The method of completing the square can e used to solve an quadratic equation. To solve a quadratic equation completing the square, ou must write the equation in the form + d. COMMON ERROR When completing the square to solve an equation, e sure to add ( ) to each side of the equation. Solving a Quadratic Equation: + d Solve completing the square ( 8) 15 + ( 8) Write the equation. Complete the square adding ( 16 ), or ( 8), to each side. ( 8) 49 Write the left side as the square of a inomial. 8 ±7 8 ± 7 Take the square root of each side. Add 8 to each side. The solutions are and Check Original equation (15)? 15 Sustitute. 1 16(1)? Simplif Solving a Quadratic Equation: a + + c 0 COMMON ERROR Before ou complete the square, e sure that the coefficient of the -term is 1. Solve completing the square Write the equation. Add 8 to each side Divide each side Complete the square adding ( 10 ), or 5, to each side. ( + 5) 9 Write the left side as the square of a inomial. + 5 ± 9 5 ± 9 Take the square root of each side. Sutract 5 from each side. The solutions are and Monitoring Progress Help in English and Spanish at BigIdeasMath.com Solve the equation completing the square. Round our solutions to the nearest hundredth, if necessar m + 1m g 4g Section 9.4 Solving Quadratic Equations Completing the Square 507

4 Finding and Using Maimum and Minimum Values One wa to find the maimum or minimum value of a quadratic function is to write the function in verte form completing the square. Recall that the verte form of a quadratic function is a( h) + k, where a 0. The verte of the graph is (h, k). Check Minimum X- 10 Y Finding a Minimum Value Find the minimum value of Write the function in verte form Write the function. Add 1 to each side Complete the square for ( + ) ( + ) 5 Simplif the left side. Write the right side as the square of a inomial. Write in verte form. The verte is (, 5). Because a is positive (a 1), the paraola opens up and the -coordinate of the verte is the minimum value. So, the function has a minimum value of 5. Finding a Maimum Value Find the maimum value of STUDY TIP Adding 1 inside the parentheses results in sutracting 1 from the right side of the equation. Write the function in verte form Write the function. Sutract 7 from each side. 7 ( ) Factor out ( + 1) Complete the square for. 8 ( + 1) 8 ( 1) ( 1) + 8 Simplif the left side. Write + 1 as the square of a inomial. Write in verte form. The verte is (1, 8). Because a is negative (a 1), the paraola opens down and the -coordinate of the verte is the maimum value. So, the function has a maimum value of 8. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Determine whether the quadratic function has a maimum or minimum value. Then find the value Chapter 9 Solving Quadratic Equations

5 Interpreting Forms of Quadratic Functions 1 f () ( + 4) + 8 g() ( 5) + 9 m() ( 3)( 1) p() ( )( 8) Which of the functions could e represented the graph? Eplain. You do not know the scale of either ais. To eliminate functions, consider the characteristics of the graph and information provided the form of each function. The graph appears to e a paraola that opens down, which means the function has a maimum value. The verte of the graph is in the first quadrant. Both -intercepts are positive. The graph of f opens down ecause a < 0, which means f has a maimum value. However, the verte ( 4, 8) of the graph of f is in the second quadrant. So, the graph does not represent f. The graph of g opens down ecause a < 0, which means g has a maimum value. The verte (5, 9) of the graph of g is in the first quadrant. B solving 0 ( 5) + 9, ou see that the -intercepts of the graph of g are and 8. So, the graph could represent g. The graph of m has two positive -intercepts. However, its graph opens up ecause a > 0, which means m has a minimum value. So, the graph does not represent m. The graph of p has two positive -intercepts, and its graph opens down ecause a < 0. This means that p has a maimum value and the verte must e in the first quadrant. So, the graph could represent p. The graph could represent function g or function p. STUDY TIP Adding 9 inside the parentheses results in sutracting 144 from the right side of the equation. Real-Life Application The function represents the height (in feet) of a model rocket seconds after it is launched. (a) Find the maimum height of the rocket. () Find and interpret the ais of smmetr. a. To find the maimum height, identif the maimum value of the function Write the function. 16( 6) Factor out ( 6 + 9) Complete the square for 6. 16( 3) Write in verte form. Because the maimum value is 144, the model rocket reaches a maimum height of 144 feet.. The verte is (3, 144). So, the ais of smmetr is 3. On the left side of 3, the height increases as time increases. On the right side of 3, the height decreases as time increases. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Determine whether the function could e represented the graph in Eample 6. Eplain. 10. h() ( 8) n() ( 5)( 0) 1. WHAT IF? Repeat Eample 7 when the function is Section 9.4 Solving Quadratic Equations Completing the Square 509

6 Solving Real-Life Prolems Modeling with Mathematics 3 ft width of order, r, ft You decide to use chalkoard paint to create a chalkoard on a door. You want the chalkoard to cover 6 square feet and to have a uniform order, as shown. Find the width of the order to the nearest inch. 7 ft chalkoard 1. Understand the Prolem You know the dimensions (in feet) of the door from the diagram. You also know the area (in square feet) of the chalkoard and that it will have a uniform order. You are asked to find the width of the order to the nearest inch.. Make a Plan Use a veral model to write an equation that represents the area of the chalkoard. Then solve the equation. 3. Solve the Prolem Let e the width (in feet) of the order, as shown in the diagram. Area of chalkoard (square feet) Length of chalkoard (feet) Width of chalkoard (feet) 6 (7 ) (3 ) 6 (7 )(3 ) Write the equation Multipl the inomials Sutract 1 from each side Divide each side Complete the square for Simplif the left side. 5 ( 5 ) Write the right side as the square of a inomial. ± 5 5 Take the square root of each side. 5 ± 5 Add 5 to each side. The solutions of the equation are and It is not possile for the width of the order to e 4.08 feet ecause the width of the door is 3 feet. So, the width of the order is aout 0.9 foot. 0.9 ft 1 in in. Convert 0.9 foot to inches. 1 ft The width of the order should e aout 11 inches. 4. Look Back When the width of the order is slightl less than 1 foot, the length of the chalkoard is slightl more than 5 feet and the width of the chalkoard is slightl more than 1 foot. Multipling these dimensions gives an area close to 6 square feet. So, an 11-inch order is reasonale. Monitoring Progress Help in English and Spanish at BigIdeasMath.com 13. WHAT IF? You want the chalkoard to cover 4 square feet. Find the width of the order to the nearest inch. 510 Chapter 9 Solving Quadratic Equations

7 9.4 Eercises Dnamic Solutions availale at BigIdeasMath.com Vocaular and Core Concept Check 1. COMPLETE THE SENTENCE The process of adding a constant c to the epression + so that + + c is a perfect square trinomial is called.. VOCABULARY Eplain how to complete the square for an epression of the form WRITING Is it more convenient to complete the square for + when is odd or when is even? Eplain. 4. WRITING Descrie how ou can use the process of completing the square to find the maimum or minimum value of a quadratic function. Monitoring Progress and Modeling with Mathematics In Eercises 5 10, find the value of c that completes the square c 6. + c c c c c In Eercises 11 16, complete the square for the epression. Then factor the trinomial. (See Eample 1.) 4. MODELING WITH MATHEMATICS Some sand art contains sand and water sealed in a glass case, similar to the one shown. When the art is turned upside down, the sand and water fall to create a new picture. The glass case has a depth of 1 centimeter and a volume of 768 cuic centimeters. ( 8) cm In Eercises 17, solve the equation completing the square. Round our solutions to the nearest hundredth, if necessar. (See Eample.) MODELING WITH MATHEMATICS The area of the patio is 16 square feet. a. Write an equation that represents the area of the patio. ( + 6) ft. Find the dimensions of the patio completing ft the square. cm a. Write an equation that represents the volume of the glass case.. Find the dimensions of the glass case completing the square. In Eercises 5 3, solve the equation completing the square. Round our solutions to the nearest hundredth, if necessar. (See Eample 3.) Section 9.4 Solving Quadratic Equations Completing the Square 511

8 33. ERROR ANALYSIS Descrie and correct the error in solving completing the square ( + 4) ± 10 4 ± f () f () In Eercises 47 50, determine whether the graph could represent the function. Eplain. 47. ( + 8)( + 3) 48. ( 5) 34. ERROR ANALYSIS Descrie and correct the error in the first two steps of solving 4 0 completing the square ( + ) ( 1)( + ) 35. NUMBER SENSE Find all values of for which is a perfect square trinomial. Eplain how ou found our answer. 36. REASONING You are completing the square to solve What is the first step? In Eercises 37 40, write the function in verte form completing the square. Then match the function with its graph A. 4 B. In Eercises 51 and 5, determine which of the functions could e represented the graph. Eplain. (See Eample 6.) 51. h() ( + ) + 3 f () ( + 3) 1 g() ( 8)( 4) m() ( + )( + 4) r() ( 5)( + 1) C. 4 D. 6 4 p() ( )( 6) q() ( + 1) In Eercises 41 46, determine whether the quadratic function has a maimum or minimum value. Then find the value. (See Eamples 4 and 5.) n() ( ) MODELING WITH MATHEMATICS The function h 16t + 48t represents the height h (in feet) of a kickall t seconds after it is kicked from the ground. (See Eample 7.) a. Find the maimum height of the kickall.. Find and interpret the ais of smmetr. 51 Chapter 9 Solving Quadratic Equations

9 54. MODELING WITH MATHEMATICS In Eercises 59 6, solve the equation completing the square. Round our solutions to the nearest hundredth, if necessar. You throw a stone from a height of 16 feet with an initial vertical velocit of 3 feet per second. The function h 16t + 3t + 16 represents the height h (in feet) of the stone after t seconds ft a. Find the maimum height of the stone PROBLEM SOLVING The distance d (in feet) that it. Find and interpret the ais of smmetr. 55. MODELING WITH MATHEMATICS You are uilding a rectangular rick patio surrounded a crushed stone order with a uniform width, as shown. You purchase patio ricks to cover 140 square feet. Find the width of the order. (See Eample 8.) ft ft ft 16 ft takes a car to come to a complete stop can e modeled d 0.05s +.s, where s is the speed of the car (in miles per hour). A car has 168 feet to come to a complete stop. Find the maimum speed at which the car can travel. 64. PROBLEM SOLVING During a ig air competition, snowoarders launch themselves from a half-pipe, perform tricks in the air, and land ack in the half-pipe. The height h (in feet) of a snowoarder aove the ottom of the half-pipe can e modeled h 16t + 4t , where t is the time (in seconds) after the snowoarder launches into the air. The snowoarder lands 3. feet lower than the height of the launch. How long is the snowoarder in the air? Round our answer to the nearest tenth of a second. ft 0 ft Initial vertical velocit 4 ft/sec 56. MODELING WITH MATHEMATICS You are making a poster that will have a uniform order, as shown. The total area of the poster is 7 square inches. Find the width of the order to the nearest inch. in. in. in. 8 in. MATHEMATICAL CONNECTIONS In Eercises 57 and 58, find the value of. Round our answer to the nearest hundredth, if necessar. 57. A 108 m 3 in. ( + 10) in. Section 9.4 hsn_alg1_pe_0904.indd 513 Cross section of a half-pipe 65. PROBLEM SOLVING You have 80 feet of fencing to make a rectangular horse pasture that covers 750 square feet. A arn will e used as one side of the pasture, as shown. w w 58. A 88 in. m ( + 6) m 16.4 ft a. Write equations for the amount of fencing to e used and the area enclosed the fencing.. Use sustitution to solve the sstem of equations from part (a). What are the possile dimensions of the pasture? Solving Quadratic Equations Completing the Square 513 /5/15 9:00 AM

10 66. HOW DO YOU SEE IT? The graph represents the quadratic function MAKING AN ARGUMENT You purchase stock for $16 per share. You sell the stock 30 das later for $3.50 per share. The price (in dollars) of a share during the 30-da period can e modeled , where is the numer of das after the stock is purchased. Your friend sas ou could have sold the stock earlier for $3.50 per share. Is our friend correct? Eplain. 4 6 a. Use the graph to estimate the -values for which 3.. Eplain how ou can use the method of completing the square to check our estimates in part (a). 67. COMPARING METHODS Consider the quadratic equation a. Solve the equation graphing.. Solve the equation completing the square. c. Compare the two methods. Which do ou prefer? Eplain. 68. THOUGHT PROVOKING Sketch the graph of the equation + 0. Identif the graph. 69. REASONING The product of two consecutive even integers that are positive is 48. Write and solve an equation to find the integers. 7. REASONING You are solving the equation What are the advantages of solving the equation completing the square instead of using other methods ou have learned? 73. PROBLEM SOLVING You are knitting a rectangular scarf. The pattern results in a scarf that is 60 inches long and 4 inches wide. However, ou have enough arn to knit 396 square inches. You decide to increase the dimensions of the scarf so that ou will use all our arn. The increase in the length is three times the increase in the width. What are the dimensions of our scarf? 70. REASONING The product of two consecutive odd integers that are negative is 195. Write and solve an equation to find the integers. Maintaining Mathematical Proficienc Write a recursive rule for the sequence. (Section 6.7) 75. a n (4, 5) (3, 0) (, 15) (1, 10) n 76. a n 4 16 (4, 4) (3, 1) 8 (, 6) (1, 3) n Simplif the epression 4ac for the given values. (Section 9.1) 74. WRITING How man solutions does + c have when c < ( )? Eplain. Reviewing what ou learned in previous grades and lessons 77. a n 0 4 n (4, 8) 8 (3, 1) 16 (, 16) (1, 0) 78. a 3, 6, c 79. a, 4, c a 1, 6, c Chapter 9 Solving Quadratic Equations