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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 (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 (x – 3)(x – 3) = x 2 – 2(3)x + 32 = x 2 – 6x (2x – 1)(2x – 1) = (2x) 2 – 2(1)(2x) = 4x 2 – 4x (x + 4)(x + 4) – 3 = x 2 + 2(4)x – 3 = x 2 + 8x + 16 – 3 = x 2 + 8x ± 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 = Quick Check

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Find the missing value to complete the square: x x +. ALGEBRA 2 LESSON 5-7 Completing the Square x x + 100Complete the square. = = 100Find. Substitute 20 for b. b2b2 b2b 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 b2b

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(continued) ALGEBRA 2 LESSON 5-7 Completing the Square Check: (–2) 2 – 2(–2 3) + ( 3) 2 – 8 – (–2 + 3) 2 + 4(–2 + 3) + 1 x 2 + 4x + 1 0=00=0 4 – – (4 + 3 – 8 + 1) + (– ) (–2) 2 + 2(–2 3) + ( 3) 2 + (–8) (–2 – 3) 2 + 4(–2 – 3) + 1 x 2 + 4x + 1 0=00= – 8 – (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 b2b2 = 9Find Quick Check

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Solve 2x 2 + 7x – 1 = 0. ALGEBRA 2 LESSON 5-7 Completing the Square x 2 + x – = Divide each side by 2.x 2 + x = Rewrite so all terms containing x are on one side. x 2 + x + = Complete the square by adding to each side x + = ±Find the square root of each side. x = – ± 7474 Solve for x = Find. b2b2 2 2 x + = Factor the perfect square trinomial 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 + – = x + – Simplify. 2 = x – Factor the perfect square trinomial. 2 = x 2 + 5x – Complete the square. Add and subtract on the right side 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 r – 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 r – 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 ] – (–250) 2 Add and subtract (–250) 2 on the right side. = –(r – 250) 2 – Factor the perfect square trinomial. = –(r – 250) Simplify in vertex form. Quick Check 5-7

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1.x x +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 –2, 87 ± 5i –4, ± i 3 4 Complete the square. 5-7

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