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4-5 Completing the Square Solve each equation by using the Square Root Property. Round to the nearest hundredth if necessary. 2. SOLUTION: Factor the perfect square trinomial. Use the Square Root Property. The solution set is {0.39, 7.61}. 4. SOLUTION: Factor the perfect square trinomial. Use the Square Root Property. The solution set is {–2.77, –0.56}. eSolutions Manual - Powered by Cognero Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square. Page 1 The solution set is {0.39, 7.61}. 4-5 Completing the Square 4. SOLUTION: Factor the perfect square trinomial. Use the Square Root Property. The solution set is {–2.77, –0.56}. Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square. 6. SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it. So: c = 25 Substitute 25 for c in the trinomial. Solve each equation by completing the square. 8. SOLUTION: eSolutions Manual - Powered by Cognero Page 2 Substitute 25 for c in the trinomial. 4-5 Completing the Square Solve each equation by completing the square. 8. SOLUTION: Use the Square Root Property. The solution set is {–4, 2}. 10. SOLUTION: Use the Square Root Property. The solution set is { –0.69, 2.19}. eSolutions Manual - Powered by Cognero 12. Page 3 The solution set is {–4, 2}. 4-5 Completing the Square 10. SOLUTION: Use the Square Root Property. The solution set is { –0.69, 2.19}. 12. SOLUTION: Use the Square Root Property. The solution set is Solve each equation by using the Square Root Property. Round to the nearest hundredth if necessary. eSolutions Manual - Powered by Cognero 14. Page 4 The solution set is 4-5 Completing the Square Solve each equation by using the Square Root Property. Round to the nearest hundredth if necessary. 14. SOLUTION: Use the Square Root Property. The solution set is { –5.16, 1.16}. 16. SOLUTION: Use the Square Root Property. The solution set is {–8.24, 0.24}. 18. SOLUTION: Use the Square Root Property. eSolutions Manual - Powered by Cognero Page 5 The solution set {–8.24, 0.24}. 4-5 Completing theisSquare 18. SOLUTION: Use the Square Root Property. The solution set is {–8.24, –3.76}. 20. SOLUTION: Use the Square Root Property. The solution set is {0.5, 4.5}. 22. SOLUTION: Use the Square Root Property. The solution set is {–17, –15}. eSolutions Manual - Powered by Cognero 24. Page 6 The solution set {0.5, 4.5}. 4-5 Completing theisSquare 22. SOLUTION: Use the Square Root Property. The solution set is {–17, –15}. 24. SOLUTION: Use the Square Root Property. The solution set is {–5.5, –1.5}. Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square. 26. SOLUTION: To find the value of c, divide the coefficient of x by 2, and square the result. Substitute 16 for c in the trinomial. eSolutions Manual - Powered by Cognero 28. Page 7 The solution set {–5.5, –1.5}. 4-5 Completing theisSquare Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square. 26. SOLUTION: To find the value of c, divide the coefficient of x by 2, and square the result. Substitute 16 for c in the trinomial. 28. SOLUTION: To find the value of c, divide the coefficient of x by 2, and square it. Substitute in the trinomial. Solve each equation by completing the square. 30. SOLUTION: Use the Square Root Property. eSolutions Manual - Powered by Cognero ; Page 8 4-5 Completing the Square Solve each equation by completing the square. 30. SOLUTION: Use the Square Root Property. ; The solution set is . 32. SOLUTION: Use the Square Root Property. The solution set is {– 4, – 2} . 34. SOLUTION: eSolutions Manual - Powered by Cognero Page 9 The solution set is {– 4, – 2} 4-5 Completing the Square . 34. SOLUTION: Use the Square Root Property. The solution set is . 36. SOLUTION: The solution set is . 38. SOLUTION: eSolutions Manual - Powered by Cognero Page 10 The solution set is 4-5 Completing the Square . 38. SOLUTION: The solution set is . 40. SOLUTION: The solution set is . 42. SOLUTION: The solution set is . 44. eSolutions Manual - Powered by Cognero SOLUTION: Page 11 The solution set 4-5 Completing theisSquare . 44. SOLUTION: ] The solution set is {–0.71, 3.11}. 46. SOLUTION: The solution set is {–1.39, 1.59}. 48. CCSS MODELING An architect’s blueprints call for a dining room measuring 13 feet by 13 feet. The customer would like the dining room to be a square, but with an area of 250 square feet. How much will this add to the dimensions of the room? SOLUTION: The area of a square is given by , where s is the side length. Therefore: eSolutions Manual - Powered by Cognero Page 12 The solution set {–1.39, 1.59}. 4-5 Completing theisSquare 48. CCSS MODELING An architect’s blueprints call for a dining room measuring 13 feet by 13 feet. The customer would like the dining room to be a square, but with an area of 250 square feet. How much will this add to the dimensions of the room? SOLUTION: The area of a square is given by , where s is the side length. Therefore: Solve for x. Therefore, about 2.81 ft should be added to the dimensions of the room. Solve each equation. Round to the nearest hundredth if necessary. 50. SOLUTION: Write the equation in standard form and solve using the quadratic formula. The quadratic formula is given by: The solution set is {–2.77, –0.56}. 52. SOLUTION: Write the equation in standard form and solve using the quadratic formula. eSolutions Manual - Powered by Cognero Page 13 The solution set is {–2.77, –0.56}. 4-5 Completing the Square 52. SOLUTION: Write the equation in standard form and solve using the quadratic formula. The quadratic formula is given by: The solution set is {–1.44, 0.24}. Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square. 54. SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it. Substitute the value of c in the trinomial. 56. SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it. eSolutions Manual - Powered by Cognero Page 14 Substitute the value of c in the trinomial. 4-5 Completing the Square 56. SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it. Substitute the value of c in the trinomial. 2 58. ERROR ANALYSIS Alonso and Aida are solving x + 8x – 20 = 0 by completing the square. Is either of them correct? Explain your reasoning. SOLUTION: eSolutions Manual - Powered by add Cognero Alonso; Aida did not 16 to each side; she added it only to the left side. Page 15 4-5 Completing the Square 2 58. ERROR ANALYSIS Alonso and Aida are solving x + 8x – 20 = 0 by completing the square. Is either of them correct? Explain your reasoning. SOLUTION: Alonso; Aida did not add 16 to each side; she added it only to the left side. 60. CCSS ARGUMENTS Without solving, determine how many unique solutions there are for each equation. Are they rational, real, or complex? Justify your reasoning. a. (x + 2)2 = 16 2 b. (x - 2) = 16 c. -(x - 2)2 = 16 2 d. 36 - (x - 2) = 16 e . 16(x + 2)2 = 0 2 2 f. (x + 4) = (x + 6) eSolutions Manual - Powered by Cognero SOLUTION: a. 2; rational; 16 is a perfect square so x + 2 and x are rational. Page 16 SOLUTION: Alonso; Aida did not add 16 to each side; she added it only to the left side. 4-5 Completing the Square 60. CCSS ARGUMENTS Without solving, determine how many unique solutions there are for each equation. Are they rational, real, or complex? Justify your reasoning. a. (x + 2)2 = 16 2 b. (x - 2) = 16 c. -(x - 2)2 = 16 2 d. 36 - (x - 2) = 16 e . 16(x + 2)2 = 0 2 2 f. (x + 4) = (x + 6) SOLUTION: a. 2; rational; 16 is a perfect square so x + 2 and x are rational. b. 2; rational; 16 is a perfect square so x – 2 and x are rational. c. 2; complex; If the opposite of square is positive, the square is negative. The square root of a negative number is complex. d. 2; real; The square must equal 20. Since that is positive but not a perfect square, the solutions will be real but not rational. e . 1; rational; The expression must be equal to 0 and only -2 makes the expression equal to 0. f. 1; rational; The expressions (x + 4) and (x + 6) must either be equal or opposites. No value makes them equal, -5 makes them opposites. The only solution is -5. 62. WRITING IN MATH Explain what it means to complete the square. Include a description of the steps you would take. SOLUTION: Completing the square allows you to rewrite one side of a quadratic equation in the form of a perfect square. Once in this form, the equation can be solved by using the Square Root Property. 64. GEOMETRY Find the area of the shaded region. Manual - Powered by Cognero eSolutions 2 F 14 m Page 17 SOLUTION: Completing the square allows you to rewrite one side of a quadratic equation in the form of a perfect square. Once in this form, the 4-5 Completing theequation Square can be solved by using the Square Root Property. 64. GEOMETRY Find the area of the shaded region. F 14 m2 2 G 18 m H 42 m2 2 J 60 m SOLUTION: The area of the outer rectangle is given by: The area of the inner rectangle is given by: The area of the shaded region is . The correct choice is H. 2 66. If 5 – 3i is a solution for x + ax + b = 0, where a and b are real numbers, what is the value of b? A 10 B 14 C 34 D 40 eSolutions Manual - Powered by Cognero SOLUTION: Substitute 5 – 3i for x in the equation. Page 18 The correct choice is H. 4-5 Completing the Square 2 66. If 5 – 3i is a solution for x + ax + b = 0, where a and b are real numbers, what is the value of b? A 10 B 14 C 34 D 40 SOLUTION: Substitute 5 – 3i for x in the equation. Equate the real and the imaginary parts. And: Substitute a = –10. The correct choice is C. Simplify. 68. SOLUTION: Write a quadratic equation in standard form with the given root(s). 70. SOLUTION: eSolutions Manual - Powered by Cognero Page 19 SOLUTION: 4-5 Completing the Square Write a quadratic equation in standard form with the given root(s). 70. SOLUTION: 72. SOLUTION: 74. SHOPPING Main St. Media sells all DVDs for one price and all books for another price. Alex bought 4 DVDs and 6 books for $170, while Matt bought 3 DVDs and 8 books for $180. What is the cost of a DVD and the cost of a book? SOLUTION: Let x and y are the cost of a DVD and a book. Multiply the first and the second equation by 3 and –4 respectively then add. Substitute 15 for y in the first equation and solve for x. eSolutions Manual - Powered by Cognero Page 20 4-5 Completing the Square 74. SHOPPING Main St. Media sells all DVDs for one price and all books for another price. Alex bought 4 DVDs and 6 books for $170, while Matt bought 3 DVDs and 8 books for $180. What is the cost of a DVD and the cost of a book? SOLUTION: Let x and y are the cost of a DVD and a book. Multiply the first and the second equation by 3 and –4 respectively then add. Substitute 15 for y in the first equation and solve for x. The cost of a DVD is $20. The cost of a book is $15. Graph each inequality. 76. SOLUTION: The boundary of the graph is the graph of 2x – 3y = 6. Since the inequality symbol is <, the line will be dashed. Test the point (0, 0). Shade the region that contains (0, 0). The graph of the inequality is: eSolutions Manual - Powered by Cognero Page 21 The cost of a DVD is $20. The cost of athe book is $15. 4-5 Completing Square Graph each inequality. 76. SOLUTION: The boundary of the graph is the graph of 2x – 3y = 6. Since the inequality symbol is <, the line will be dashed. Test the point (0, 0). Shade the region that contains (0, 0). The graph of the inequality is: Write the piecewise function shown in each graph. 78. SOLUTION: The left portion is the graph of the constant function f (x) = –4. There is a dot at –2, so the constant function is defined for x ≤ –2. The center portion is the graph of f (x) = –2x + 3. There are circles at –2 and 1, so the function is defined for –2 < x < 1. The right portion is the graph of f (x) = x – 5. There is a dot at 2, so the function is defined for x ≥ 2. The piece-wise function is: eSolutions Manual - Powered by Cognero Page 22 4-5 Completing the Square Write the piecewise function shown in each graph. 78. SOLUTION: The left portion is the graph of the constant function f (x) = –4. There is a dot at –2, so the constant function is defined for x ≤ –2. The center portion is the graph of f (x) = –2x + 3. There are circles at –2 and 1, so the function is defined for –2 < x < 1. The right portion is the graph of f (x) = x – 5. There is a dot at 2, so the function is defined for x ≥ 2. The piece-wise function is: 80. SOLUTION: The left portion is the graph of f (x) = x + 12. There is dot at –6, so the function is defined for x ≤ –6. The center portion is the graph of the constant function f (x) = 8. There are circles at –6 and 2, so the constant function is defined for –6 < x < 2. The right portion is the graph of f (x) = –2.5x + 15. There is a dot at 2, so the function is defined for x ≥ 2. The piece-wise function is: 2 Evaluate b – 4ac for the given values of a, b, and c. eSolutions Manual - Powered by Cognero 82. a = –2, b = –7, c = 3 Page 23 The piece-wise function is: 4-5 Completing the Square 2 Evaluate b – 4ac for the given values of a, b, and c. 82. a = –2, b = –7, c = 3 SOLUTION: Substitute a = –2, b = –7, and c = 3. eSolutions Manual - Powered by Cognero Page 24