3.5 Completing the Square Essential Question How can you complete the square for a quadratic epression? Using Algera Tiles to Complete the Square Work with a partner. Use algera tiles to complete the square for the epression + 6. a. You can model + 6 using one -tile and si -tiles. Arrange the tiles in a square. Your arrangement will e incomplete in one of the corners.. How many 1-tiles do you need to complete the square? c. Find the value of c so that the epression + 6 + c is a perfect square trinomial. d. Write the epression in part (c) as the square of a inomial. Drawing Conclusions Work with a partner. a. Use the method outlined in Eploration 1 to complete the tale. LOOKING FOR STRUCTURE To e proficient in math, you need to look closely to discern a pattern or structure. Epression + + c + 4 + c + 8 + c + 10 + c Value of c needed to complete the square Epression written as a inomial squared. Look for patterns in the last column of the tale. Consider the general statement + + c = ( + d ). How are d and related in each case? How are c and d related in each case? c. How can you otain the values in the second column directly from the coefficients of in the first column? Communicate Your Answer 3. How can you complete the square for a quadratic epression? 4. Descrie how you can solve the quadratic equation + 6 = 1 y completing the square. Section 3.5 Completing the Square 135
3.5 Lesson Core Vocaulary completing the square, p. 136 Previous perfect square trinomial verte form ANOTHER WAY You can also solve the equation y writing it in standard form as 16 36 = 0 and factoring. What You Will Learn Solve quadratic equations using square roots. Solve quadratic equations y completing the square. Write quadratic functions in verte form. Solving Quadratic Equations Using Square Roots Previously, you have solved equations of the form u = d y taking the square root of each side. This method also works when one side of an equation is a perfect square trinomial and the other side is a constant. Solving a Quadratic Equation Using Square Roots Solve 16 + 64 = 100 using square roots. 16 + 64 = 100 Write the equation. ( 8) = 100 Write the left side as a inomial squared. 8 = ±10 = 8 ± 10 Take square root of each side. Add 8 to each side. So, the solutions are = 8 + 10 = 18 and = 8 10 =. Solve the equation using square roots. Check your solution(s). 1. + 4 + 4 = 36. 6 + 9 = 1 3. + 11 = 81 In Eample 1, the epression 16 + 64 is a perfect square trinomial ecause it equals ( 8). Sometimes you need to add a term to an epression + to make it a perfect square trinomial. This process is called completing the square. Core Concept Completing the Square Words To complete the square for the epression +, add ( Diagrams In each diagram, the comined area of the shaded regions is +. Adding ( ) ) completes the square in the second diagram. ( (. ( ( ( ( Algera + + ( = ) ( + ) ( + ) ( = + ) 136 Chapter 3 Quadratic Equations and Comple Numers
Solving Quadratic Equations y Completing the Square Making a Perfect Square Trinomial Find the value of c that makes + 14 + c a perfect square trinomial. Then write the epression as the square of a inomial. 14 Step 1 Find half the coefficient of. = 7 Step Square the result of Step 1. 7 = 49 7 7 Step 3 Replace c with the result of Step. + 14 + 49 7 7 49 The epression + 14 + c is a perfect square trinomial when c = 49. Then + 14 + 49 = ( + 7)( + 7) = ( + 7). Find the value of c that makes the epression a perfect square trinomial. Then write the epression as the square of a inomial. 4. + 8 + c 5. + c 6. 9 + c LOOKING FOR STRUCTURE Notice you cannot solve the equation y factoring ecause 10 + 7 is not factorale as a product of inomials. The method of completing the square can e used to solve any quadratic equation. When you complete the square as part of solving an equation, you must add the same numer to oth sides of the equation. Solving a + + c = 0 when a = 1 Solve 10 + 7 = 0 y completing the square. 10 + 7 = 0 Write the equation. 10 = 7 Write left side in the form +. 10 + 5 = 7 + 5 Add ( ) = ( 10 ) = 5 to each side. ( 5) = 18 Write left side as a inomial squared. 5 = ± 18 = 5 ± 18 = 5 ± 3 Take square root of each side. Add 5 to each side. Simplify radical. The solutions are = 5 + 3 and = 5 3. You can check this y graphing y = 10 + 7. The -intercepts are aout 9.4 5 + 3 and 0.76 5 3. Check 8 1 Zero X=9.46407 Y=0 4 Section 3.5 Completing the Square 137
Solving a + + c = 0 when a 1 Solve 3 + 1 + 15 = 0 y completing the square. The coefficient a is not 1, so you must first divide each side of the equation y a. 3 + 1 + 15 = 0 Write the equation. + 4 + 5 = 0 Divide each side y 3. + 4 = 5 Write left side in the form +. + 4 + 4 = 5 + 4 Add ( ) = ( ) 4 = 4 to each side. ( + ) = 1 Write left side as a inomial squared. + = ± 1 = ± 1 Take square root of each side. Sutract from each side. = ± i Write in terms of i. The solutions are = + i and = i. Solve the equation y completing the square. 7. 4 + 8 = 0 8. + 8 5 = 0 9. 3 18 6 = 0 10. 4 + 3 = 68 11. 6( + ) = 4 1. ( ) = 00 Writing Quadratic Functions in Verte Form Recall that the verte form of a quadratic function is y = a( h) + k, where (h, k) is the verte of the graph of the function. You can write a quadratic function in verte form y completing the square. Writing a Quadratic Function in Verte Form Check 1 4 Minimum X=6 Y=-18 6 1 Write y = 1 + 18 in verte form. Then identify the verte. y = 1 + 18 y +? = ( 1 +?) + 18 y + 36 = ( 1 + 36) + 18 Add ( ) y + 36 = ( 6) + 18 Write the function. y = ( 6) 18 Solve for y. Prepare to complete the square. = ( 1 ) = 36 to each side. Write 1 + 36 as a inomial squared. The verte form of the function is y = ( 6) 18. The verte is (6, 18). Write the quadratic function in verte form. Then identify the verte. 13. y = 8 + 18 14. y = + 6 + 4 15. y = 6 138 Chapter 3 Quadratic Equations and Comple Numers
Modeling with Mathematics The height y (in feet) of a aseall t seconds after it is hit can e modeled y the function y = 16t + 96t + 3. Find the maimum height of the aseall. How long does the all take to hit the ground? ANOTHER WAY You can use the coefficients of the original function y = f () to find the maimum height. f ( a) ( = f 96 ( 16)) = f (3) = 147 LOOKING FOR STRUCTURE You could write the zeros as 3 ± 7 3, ut it is 4 easier to recognize that 3 9.1875 is negative ecause 9.1875 is greater than 3. 1. Understand the Prolem You are given a quadratic function that represents the height of a all. You are asked to determine the maimum height of the all and how long it is in the air.. Make a Plan Write the function in verte form to identify the maimum height. Then find and interpret the zeros to determine how long the all takes to hit the ground. 3. Solve the Prolem Write the function in verte form y completing the square. y = 16t + 96t + 3 Write the function. y = 16(t 6t) + 3 y +? = 16(t 6t +?) + 3 y + ( 16)(9) = 16(t 6t + 9) + 3 y 144 = 16(t 3) + 3 y = 16(t 3) + 147 Solve for y. Factor 16 from first two terms. Prepare to complete the square. Add ( 16)(9) to each side. The verte is (3, 147). Find the zeros of the function. 0 = 16(t 3) + 147 Sustitute 0 for y. 147 = 16(t 3) Write t 6t + 9 as a inomial squared. Sutract 147 from each side. 9.1875 = (t 3) Divide each side y 16. ± 9.1875 = t 3 Take square root of each side. 3 ± 9.1875 = t Add 3 to each side. Reject the negative solution, 3 9.1875 0.03, ecause time must e positive. So, the maimum height of the all is 147 feet, and it takes 3 + 9.1875 6 seconds for the all to hit the ground. 4. Look Back The verte indicates that the maimum height of 147 feet occurs when t = 3. This makes sense ecause the graph of the function is paraolic with zeros near t = 0 and t = 6. You can use a graph to check the maimum height. Maimum 0 X=3 Y=147 0 16. WHAT IF? The height of the aseall can e modeled y y = 16t + 80t +. Find the maimum height of the aseall. How long does the all take to hit the ground? 180 7 Section 3.5 Completing the Square 139
3.5 Eercises Dynamic Solutions availale at BigIdeasMath.com Vocaulary and Core Concept Check 1. VOCABULARY What must you add to the epression + to complete the square?. COMPLETE THE SENTENCE The trinomial 6 + 9 is a ecause it equals. and Modeling with Mathematics In Eercises 3 10, solve the equation using square roots. Check your solution(s). (See Eample 1.) 3. 8 + 16 = 5 4. r 10r + 5 = 1 5. 18 + 81 = 5 6. m + 8m + 16 = 45 7. y 4y + 144 = 100 8. 6 + 169 = 13 9. 4w + 4w + 1 = 75 10. 4 8 + 4 = 1 In Eercises 11 0, find the value of c that makes the epression a perfect square trinomial. Then write the epression as the square of a inomial. (See Eample.) 11. + 10 + c 1. + 0 + c 13. y 1y + c 14. t t + c 15. 6 + c 16. + 4 + c 17. z 5z + c 18. + 9 + c 19. w + 13w + c 0. s 6s + c In Eercises 1 4, find the value of c. Then write an epression represented y the diagram. 1. 3.. c 6 6 8 8 4. 8 8 c 10 10 In Eercises 5 36, solve the equation y completing the square. (See Eamples 3 and 4.) 5. + 6 + 3 = 0 6. s + s 6 = 0 7. + 4 = 0 8. t 8t 5 = 0 9. z(z + 9) = 1 30. ( + 8) = 0 31. 7t + 8t + 56 = 0 3. 6r + 6r + 1 = 0 33. 5( + 6) = 50 34. 4w(w 3) = 4 35. 4 30 = 1 + 10 36. 3s + 8s = s 9 37. ERROR ANALYSIS Descrie and correct the error in solving the equation. 4 + 4 11 = 0 4 ( + 6 ) = 11 4 ( + 6 + 9 ) = 11 + 9 4( + 3) = 0 ( + 3) = 5 + 3 = ± 5 = 3 ± 5 38. ERROR ANALYSIS Descrie and correct the error in finding the value of c that makes the epression a perfect square trinomial. + 30 + c + 30 + 30 + 30 + 15 6 6 c 10 10 c 39. WRITING Can you solve an equation y completing the square when the equation has two imaginary solutions? Eplain. 140 Chapter 3 Quadratic Equations and Comple Numers
40. ABSTRACT REASONING Which of the following are solutions of the equation a + a =? Justify your answers. A a B a C D a E a F a + USING STRUCTURE In Eercises 41 50, determine whether you would use factoring, square roots, or completing the square to solve the equation. Eplain your reasoning. Then solve the equation. 41. 4 1 = 0 4. + 13 + = 0 43. ( + 4) = 16 44. ( 7) = 9 45. + 1 + 36 = 0 46. 16 + 64 = 0 47. + 4 3 = 0 48. 3 + 1 + 1 = 0 60. f() = + 6 16 61. f() = 3 + 4 6. g() = + 7 + 63. MODELING WITH MATHEMATICS While marching, a drum major tosses a aton into the air and catches it. The height h (in feet) of the aton t seconds after it is thrown can e modeled y the function h = 16t + 3t + 6. (See Eample 6.) a. Find the maimum height of the aton.. The drum major catches the aton when it is 4 feet aove the ground. How long is the aton in the air? 64. MODELING WITH MATHEMATICS A firework eplodes when it reaches its maimum height. The height h (in feet) of the firework t seconds after it is launched can e modeled y h = 500 9 t + 1000 3 t + 10. What is the maimum height of the firework? How long is the firework in the air efore it eplodes? 49. 100 = 0 50. 4 0 = 0 MATHEMATICAL CONNECTIONS In Eercises 51 54, find the value of. 51. Area of 5. Area of rectangle = 50 parallelogram = 48 + 10 + 6 53. Area of triangle = 40 54. Area of trapezoid = 0 + 4 3 1 + 9 In Eercises 55 6, write the quadratic function in verte form. Then identify the verte. (See Eample 5.) 55. f() = 8 + 19 56. g() = 4 1 57. g() = + 1 + 37 58. h() = + 0 + 90 59. h() = + 48 65. COMPARING METHODS A skateoard shop sells aout 50 skateoards per week when the advertised price is charged. For each $1 decrease in price, one additional skateoard per week is sold. The shop s revenue can e modeled y y = (70 )(50 + ). SKATEBOARDS Quality Skateoards for $70 a. Use the intercept form of the function to find the maimum weekly revenue.. Write the function in verte form to find the maimum weekly revenue. c. Which way do you prefer? Eplain your reasoning. Section 3.5 Completing the Square 141
66. HOW DO YOU SEE IT? The graph of the function f () = ( h) is shown. What is the -intercept? Eplain your reasoning. f y (0, 9) 69. MAKING AN ARGUMENT Your friend says the equation + 10 = 0 can e solved y either completing the square or factoring. Is your friend correct? Eplain. 70. THOUGHT PROVOKING Write a function g in standard form whose graph has the same -intercepts as the graph of f () = + 8 +. Find the zeros of each function y completing the square. Graph each function. 71. CRITICAL THINKING Solve + + c = 0 y completing the square. Your answer will e an epression for in terms of and c. 67. WRITING At Buckingham Fountain in Chicago, the height h (in feet) of the water aove the main nozzle can e modeled y h = 16t + 89.6t, where t is the time (in seconds) since the water has left the nozzle. Descrie three different ways you could find the maimum height the water reaches. Then choose a method and find the maimum height of the water. 68. PROBLEM SOLVING A farmer is uilding a rectangular pen along the side of a arn for animals. The arn will serve as one side of the pen. The farmer has 10 feet of fence to enclose an area of 151 square feet and wants each side of the pen to e at least 0 feet long. a. Write an equation that represents the area of the pen.. Solve the equation in part (a) to find the dimensions of the pen. 10 7. DRAWING CONCLUSIONS In this eercise, you will investigate the graphical effect of completing the square. a. Graph each pair of functions in the same coordinate plane. y = + y = 6 y = ( + 1) y = ( 3). Compare the graphs of y = + and y = ( + ). Descrie what happens to the graph of y = + when you complete the square. 73. MODELING WITH MATHEMATICS In your pottery class, you are given a lump of clay with a volume of 00 cuic centimeters and are asked to make a cylindrical pencil holder. The pencil holder should e 9 centimeters high and have an inner radius of 3 centimeters. What thickness should your pencil holder have if you want to use all of the clay? Top view 3 cm cm 3 cm cm 9 cm cm cm Side view Maintaining Mathematical Proficiency Solve the inequality. Graph the solution. 74. 3 < 5 75. 4 8y 1 76. n 3 + 6 > 1 77. s 5 8 Graph the function. Lael the verte, ais of symmetry, and -intercepts. 78. g() = 6( 4) 79. h() = ( 3) 80. f () = + + 5 81. f () = ( + 10)( 1) Reviewing what you learned in previous grades and lessons 14 Chapter 3 Quadratic Equations and Comple Numers