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Section 5.7 Completing the Square

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(For help, go to Lessons 5-1 and page 262.) ALGEBRA 2 LESSON 5-7 Completing the Square 5-7 9 16 1.(x – 3)(x – 3)2.(2x – 1)(2x – 1) 3.(x + 4)(x + 4) – 34.± 25 5.± 486.± –4 7.± Simplify each expression. Check Skills You’ll Need

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Solutions ALGEBRA 2 LESSON 5-7 Completing the Square 9 16 9 16 3434 1.(x – 3)(x – 3) = x 2 – 2(3)x + 32 = x 2 – 6x + 9 2.(2x – 1)(2x – 1) = (2x) 2 – 2(1)(2x) + 1 2 = 4x 2 – 4x + 1 3.(x + 4)(x + 4) – 3 = x 2 + 2(4)x + 4 2 – 3 = x 2 + 8x + 16 – 3 = x 2 + 8x + 13 4.± 25 = ±5 5.± 48 = ± 16 3 = ±4 3 6.± –4 = ± –1 4 = ±2i 7.±= ±= ± 5-7

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Solve x 2 – 12x + 36 = 9. ALGEBRA 2 LESSON 5-7 Completing the Square x 2 – 12x + 36 = 9 (x – 6) 2 = 9Factor the trinomial. x – 6 = ±3Find the square root of each side. x – 6 = 3 or x – 6 = –3Solve for x. x = 9 orx = 3 5-7 Quick Check

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Find the missing value to complete the square: x 2 + 20x +. ALGEBRA 2 LESSON 5-7 Completing the Square x 2 + 20x + 100Complete the square. = = 100Find. Substitute 20 for b. b2b2 b2b2 20 2 2 22 5-7 Quick Check

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x + 2 = ± 3Find the square root of each side. x = –2 ± 3Solve for x. Solve x 2 + 4x + 1 = 0. ALGEBRA 2 LESSON 5-7 Completing the Square (x + 2) 2 = 3Factor the perfect square trinomal. x 2 + 4x = –1Rewrite so all terms containing x are on one side. x 2 + 4x + 4 = –1 + 4Complete the square by adding 4 to each side. = 4Find. 4242 b2b2 2 2 5-7

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(continued) ALGEBRA 2 LESSON 5-7 Completing the Square Check:0 0 0 0 0 0 0 0 0 0 (–2) 2 – 2(–2 3) + ( 3) 2 – 8 – 4 3 + 1 (–2 + 3) 2 + 4(–2 + 3) + 1 x 2 + 4x + 1 0=00=0 4 – 4 3 + 3 – 8 + 4 3 + 1 (4 + 3 – 8 + 1) + (–4 3 + 4 3) (–2) 2 + 2(–2 3) + ( 3) 2 + (–8) + 4 3 + 1 (–2 – 3) 2 + 4(–2 – 3) + 1 x 2 + 4x + 1 0=00=0 4 + 4 3 + 3 – 8 – 4 3 + 1 (4 + 3 – 8 + 1) + (4 3 – 4 3) 5-7 Quick Check

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Solve x 2 + 6x + 12 = 0. ALGEBRA 2 LESSON 5-7 Completing the Square x 2 + 6x = –12Rewrite so all terms containing x are on one side. x 2 + 6x + 9 = –12 + 9Complete the square by adding 9 to each side. (x + 3) 2 = –3Factor the perfect square trinomial. x + 3 = ± –3 Find the square root of each side. x = –3 ± –3 Solve for x. = –3 ± i 3 Simplify. 6262 b2b2 = 9Find. 2 2 5-7 Quick Check

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Solve 2x 2 + 7x – 1 = 0. ALGEBRA 2 LESSON 5-7 Completing the Square x 2 + x – = 0 7272 1212 Divide each side by 2.x 2 + x = 1212 7272 Rewrite so all terms containing x are on one side. x 2 + x + =+ 7272 49 16 49 16 1212 Complete the square by adding to each side. 49 16 7474 57 4 x + = ±Find the square root of each side. x = – ± 7474 Solve for x. 57 4 = 7272 2 49 16 Find. b2b2 2 2 x + = 7474 57 16 Factor the perfect square trinomial. 2 5-7 Quick Check

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Write y = x 2 + 5x + 2 in vertex form. ALGEBRA 2 LESSON 5-7 Completing the Square y = x 2 + 5x + 2 The vertex form is y = x + –. 5252 17 4 2 = x + – 5252 17 4 Simplify. 2 = x + + 2 – 25 4 5252 Factor the perfect square trinomial. 2 = x 2 + 5x + + 2 – 5252 5252 Complete the square. Add and subtract on the right side. 5252 2 2 2 5-7 Quick Check

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The monthly profit P from the sales of rugs woven by a family rug-making business depends on the price r that they charge for a rug. The profit is model by P = –r 2 + 500r – 59,500. Write the function in vertex form. Use the vertex form to find the price that yields the maximum monthly profit and the amount of the maximum profit. ALGEBRA 2 LESSON 5-7 Completing the Square P = –r 2 + 500r – 59500 The vertex is (250, 3000). A price of $250 per rug gives a maximum monthly profit of $3000. = –(r 2 – 500r) – 59500Factor –1 from the first two terms. = –[r 2 – 500r + (–250) 2 ] – 59500 + (–250) 2 Add and subtract (–250) 2 on the right side. = –(r – 250) 2 – 59500 + 62500Factor the perfect square trinomial. = –(r – 250) 2 + 3000Simplify in vertex form. Quick Check 5-7

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1.x 2 + 60x +2.x 2 – 7x + Simplify each expression. 3.x 2 – 6x – 16 = 04.x 2 – 14x + 74 = 0 5.3x 2 + 5x – 28 = 06.4x 2 – 6x + 3 = 0 ALGEBRA 2 LESSON 5-7 Completing the Square 900 49 9 –2, 87 ± 5i –4, 7373 3434 ± i 3 4 Complete the square. 5-7

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