4-6 Completing the Square Content Standard Reviews A.REI.4. Solve quadratic equations y... completing the square... Ojectives To solve equations y completing the square To rewrite functions y completing the square Can you write the area of your square in two ways? How can you use pieces like these to form a square with side length 1 3 (and no overlapping pieces)? Show a sketch of your solution. How many of each piece do you need? Eplain. MATHEMATICAL PRACTICES Forming a square with model pieces provides a useful geometric image for completing a square algeraically. Lesson Vocaulary completing the square Essential Understanding Completing a perfect square trinomial allows you to factor the completed trinomial as the square of a inomial. You can solve an equation that contains a perfect square y finding square roots. The simplest of this type of equation has the form a 5 c. How is solving this equation like solving a linear equation? You isolate the variale term. Prolem 1 Solving y Finding Square Roots What is the solution of each equation? A 4 1 10 5 46 B 3 5 5 5 4 5 36 d Rewrite in a 5 c form. S 3 5 30 4 4 5 36 4 d Isolate. S 3 3 5 30 3 5 9 5 10 5 43 d Find square roots. S 5 4!10 1. What is the solution of each equation? a. 7 10 5 5. 1 9 5 13 Lesson 4-6 Completing the Square 33
Prolem Determining Dimensions D IC AC ES AM YN TIVITI Dynamic Activity Completing the Square STEM Architecture While designing a house, an architect used windows like the one shown here. What are the dimensions of the window if it has 766 square inches of glass? Step 1 Find the area of the window. The area of the rectangular part is ()() 5 in.. The area of the semicircular part is p 1 1 1 pr 5 p Q R 5 p 4 5 8 in.. So, the total amount of glass used is p 1 8 5 766 in.. Step Solve for. p Q 1 8 R 5 766 Is the answer reasonale? Yes; the rectangular part is aout 30 3 70 5 100 in.. This leaves enough glass for the semicircle. 5 766 1 p8 < 434 Write the equation in a 5 c form. Isolate. Find square roots. Use a calculator. Length cannot e negative. So the rectangular portion of the window is 34 in. wide y 68 in. long. The semicircular top has a radius of 17 in.. The lengths of the sides of a rectangular window have the ratio 1.6 to 1. The area of the window is 8.4 in.. What are the window dimensions? Sometimes an equation shows a perfect square trinomial equal to a constant. To solve, factor the perfect square trinomial into the square of a inomial. Then find square roots. Prolem 3 Solving a Perfect Square Trinomial Equation What is the solution of 1 4 1 4 5 5? Factor the perfect square trinomial. Find square roots. Rewrite as two equations. Solve for. 1 4 1 4 5 5 ( 1 ) 5 5 1 5 w5 1 5 5 or 1 5 5 5 3 or 5 7 3. What is the solution of 14 1 49 5 5? 34 Chapter 4 033_hsm11ase_cc_0406.indd 34 Quadratic Functions and Equations 3/8/11 8:5:30 PM
If 1 is not part of a perfect square trinomial, you can use the coefficient to find a constant c so that 1 1 c is a perfect square. When you do this, you are completing the square. The diagram models this process. Key Concept Completing the Square You can form a perfect square trinomial from 1 y adding Q R. 1 1 Q R 5 Q 1 R Why do you want a perfect square trinomial? You can factor a perfect square trinomial into the square of a inomial. Prolem 4 Completing the Square What value completes the square for 10? Justify your answer. 10 Identify ; 510 Q R 10 5 Q R 5 (5) 5 5 Find Q R. 10 1 5 Add the value of Q R to complete the square. 10 1 5 5 ( 5) Rewrite as the square of a inomial. 4. a. What value completes the square for 1 6?. Reasoning Is it possile for more than one value to complete the square for an epression? Eplain. Key Concept Solving an Equation y Completing the Square 1. Rewrite the equation in the form 1 5 c. To do this, get all terms with the variale on one side of the equation and the constant on the other side. Divide all the terms of the equation y the coefficient of if it is not 1.. Complete the square y adding Q R to each side of the equation. 3. Factor the trinomial. 4. Find square roots. 5. Solve for. Lesson 4-6 Completing the Square 35 m11ase_cc_0406.indd 35 3/8/11 8:
Is there a way to check without a calculator? Yes; you can check that your solutions are reasonale y estimating. Prolem 5 Solving y Completing the Square What is the solution of 3 1 1 6 5 0? 3 1 1 6 5 0 3 1 56 Rewrite. Get all terms with on one side of the equation. 3 3 1 3 5 6 3 4 5 Simplify. Q R 5 Q 4 R 5 () 5 4 Find Q R 5 4. 4 1 4 5 1 4 Divide each side y 3 so the coefficient of will e 1. Add 4 to each side. ( ) 5 Factor the trinomial. 5 4! Find square roots. 5 4! Solve for. Check your results on your calculator. Replace in the original equation with 1! and!. 5. What is the solution of 1 3 5 1 9? You can complete a square to change a quadratic function to verte form. Prolem 6 Writing in Verte Form What should e your first step? Complete the square. What is y 5 1 4 6 in verte form? Name the verte and y-intercept. y 5 1 4 6 y 5 1 4 1 6 y 5 ( 1 ) 6 4 Add Q R 5 to complete the square. Also, sutract to leave the function unchanged. Factor the perfect square trinomial. y 5 ( 1 ) 10 Simplify. The verte is (, 10). The y-intercept is (0, 6). Check with a graphing calculator. Plot1 Plot Plot3 X \Y1 = X +4X 6 4 \Y = 3 \Y3 = 1 \Y4 = 0 \Y5 = 1 \Y6 = Minimum \Y7 = X= Y= 10 Y1= 10 Y1 6 9 10 9 6 1 6 6. What is y 5 1 3 6 in verte form? Name the verte and y-intercept. 36 Chapter 4 Quadratic Functions and Equations m11ase_cc_0406.indd 36 4/8/11 5:
Lesson Check Do you know HOW? Solve each equation y finding square roots. 1. 5 7. 6 5 54 Complete the square. Do you UNDERSTAND? MATHEMATICAL PRACTICES 9. Vocaulary Eplain the process of completing the square. 10. How can you rewrite the equation 1 1 1 5 5 3 so the left side of the equation is in the form ( 1 a)? 3. 1 1 j 4. 1 10 1 j 5. 4 1 j 6. 1 1 1 j 7. 1 100 1 j 8. 3 1 j 11. Error Analysis Your friend completed the square and wrote the epression shown. Eplain your friend s error and write the epression correctly. Practice and Prolem-Solving Eercises MATHEMATICAL PRACTICES A Practice Solve each equation y finding square roots. See Prolem 1. 1. 5 5 80 13. 4 5 0 14. 5 3 15. 9 5 5 16. 3 15 5 0 17. 5 40 5 0 18. Fitness A rectangular swimming pool is 6 ft deep. One side of the pool is.5 times longer than the other. The amount of water needed to fill the swimming pool is 160 cuic feet. Find the dimensions of the pool. Solve each equation. See Prolem. See Prolem 3. 19. 1 6 1 9 5 1 0. 4 1 4 5 100 1. 1 1 5 4. 1 8 1 16 5 16 9 3. 4 1 4 1 1 5 49 4. 1 1 36 5 5 5. 5 1 10 1 1 5 9 6. 30 1 5 5 400 7. 9 1 4 1 16 5 36 Complete the square. See Prolem 4. 8. 1 18 1 j 9. 1 j 30. 4 1 j 31. 1 0 1 j 3. m 3m 1 j 33. 1 4 1 j Solve each quadratic equation y completing the square. See Prolem 5. 34. 1 6 3 5 0 35. 1 1 7 5 0 36. 1 4 1 5 0 37. 5 5 38. 1 8 5 11 39. 1 1 5 10 40. 3 5 1 41. 1 5 6 1 4 4. 1 5 5 43. 4 1 10 3 5 0 44. 9 1 5 0 45. 5 1 30 5 1 Lesson 4-6 Completing the Square 37 m11ase_cc_0406.indd 37 3/8/11 8:
Rewrite each equation in verte form. See Prolem 6. 46. y 5 1 4 1 1 47. y 5 8 1 1 48. y 5 1 3 49. y 5 1 4 7 50. y 5 6 1 51. y 5 1 4 1 B Apply 5. Think Aout a Plan Th e area of the rectangle shown is 80 square inches. What is the value of? How can you write an equation to represent 80 in terms of? How can you find the value of y completing the square? 3 Find the value of k that would make the left side of each equation a perfect square trinomial. 53. 1 k 1 5 5 0 54. k 1 100 5 0 55. k 1 11 5 0 56. 1 k 1 64 5 0 57. k 1 81 5 0 58. 5 k 1 1 5 0 59. 1 k 1 1 4 5 0 60. 9 k 1 4 5 0 61. 36 k 1 49 5 0 6. Geometry Th e tale shows some possile dimensions of rectangles with a perimeter of 100 units. Copy and complete the tale. a. Plot the points (width, area). Find a model for the data set.. What is another point in the data set? Use it to verify your model. c. What is a reasonale domain for this function? Eplain. d. Find the maimum possile area. What dimensions yield this area? e. Find a function for area in terms of width without using the tale. Do you get the same model as in part (a)? Eplain. Width 1 3 4 5 Length 49 48 Area 49 Solve each quadratic equation y completing the square. 63. 1 5 3 5 0 64. 1 3 5 65. 5 5 66. 1 1 5 0 67. 3 4 5 68. 5 5 4 69. 1 3 4 5 1 71. 3 1 5 3 70. 1 5 1 8 7. 1 1 4 5 0 73. 6 5 74. 0.5 0.6 1 0.3 5 0 75. Footall Th e quadratic function h 50.01 1 1.18 1 models the height of a punted footall. The horizontal distance in feet from the point of impact with the kicker s foot is, and h is the height of the all in feet. a. Write the function in verte form. What is the maimum height of the punt?. The nearest defensive player is 5 ft horizontally from the point of impact. How high must the player reach to lock the punt? c. Suppose the all was not locked ut continued on its path. How far down the field would the all go efore it hit the ground? 38 Chapter 4 Quadratic Functions and Equations m11ase_cc_0406.indd 38 3/8/11 8:
C Challenge Solve for in terms of a. 76. a 5 6a 77. 3 1 a 5 a 78. a 8a 56 79. 4a 1 8a 1 3 5 0 80. 3 1 a 5 9 1 9a 81. 6a 11a 5 10 8. Solve 5 (6!) 1 7 y completing the square. Rewrite each equation in verte form. Then find the verte of the graph. 83. y 54 5 1 3 84. y 5 1 5 1 1 85. y 5 1 5 1 4 5 1 11 5 Standardized Test Prep SAT/ACT 1 86. The graph of which inequality has its verte at Q, 5R? y, u 5 u 1 5 y. u 1 5 u 5 y, u 1 5 u 5 y. u 5 u 5 87. Which numer is a solution of u 9 u 5 9 1? 3 0 3 6 88. Joanne tosses an apple seed on the ground. It travels along a paraola with the equation y 5 1 4. Assume the seed was thrown from a height of 4 ft. How many feet away from Joanne will the apple seed land? 1 ft ft 4 ft 8 ft Etended Response 89. List the steps for solving the equation 9 58 y the completing the square method. Eplain each step. Mied Review Solve each equation y factoring. Check your answers. See Lesson 4-5. 90. 3 1 1 5 0 91. 4 53 9. 16 1 5 3 Determine whether a quadratic model eists for each set of values. If so, write the model. See Lesson 4-3. 93. (4, 3), (3, 3), (, 4) 94. Q1, 1 R, (0, ), (, ) 95. (0, ), (1, 0), (, 4) Solve each system y elimination. 1 y 5 4 96. e 3 y 5 6 1 y 5 7 97. e 1 5y 51 1 4y 5 10 98. e 3 1 5y 5 14 See Lesson 3-. Get Ready! To prepare for Lesson 4-7, do Eercises 99 100. Evaluate each epression for the given values of the variales. See Lesson 1-3. 99. 4ac; a 5 1, 5 6, c 5 3 100. 4ac; a 55, 5, c 5 4 Lesson 4-6 Completing the Square 39 m11ase_cc_0406.indd 39 3/8/11 8: