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Completing the Square (For help, go to Lessons 9-4 and 9-7.) Find each square. 1.(d – 4) 2 2.(x + 11) 2 3.(k – 8) 2 Factor. 4.b 2 + 10b + 255.t 2 + 14t + 496.n 2 – 18n + 81 10-5 Check Skills You’ll Need

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Completing the Square 1. (d – 4) 2 = d 2 – 2d(4) + 4 2 = d 2 – 8d + 16 2. (x + 11) 2 = x 2 + 2x(11) + 11 2 = x 2 + 22x + 121 3. (k – 8) 2 = k 2 – 2k(8) + 8 2 = k 2 – 16k + 64 4. b 2 + 10b + 25 = b 2 + 2b(5) + 5 2 = (b + 5) 2 5. t 2 + 14t + 49 = t 2 + 2t(7) + 7 2 = (t + 7) 2 6. n 2 – 18n + 81 = n 2 – 2n(9) + 9 2 = (n – 9) 2 Solutions 10-5

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Completing the Square Find the value of c to complete the square for x 2 – 16x + c. The value of b in the expression x 2 – 16x + c is –16. The term to add to x 2 – 16x isor 64. –16 2 2 Quick Check 10-5

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Completing the Square First, write the left side of the equation as a perfect square. X 2 – 4x = 12 X 2 – 4x + 4 = 12 + 4 Second, solve the equation by taking the square root of each side. (x – 2) 2 = V16 X – 2 = ±4 X = 4 + 2 and x = -4 + 2 X = 6 and -2

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Completing the Square Did you see that this can be factored using two binomials?. X 2 – 4x = 12 X 2 – 4x – 12 = 0 (x – 6)( x+ 2) = 0 X = 6 and -2

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Completing the Square Solve the equation x 2 + 5x = 50. Step 1: Write the left side of x 2 + 5x = 50 as a perfect square. x 2 + 5x = 50 x 2 + 5x + = 50 + 5252 2 5252 2 Add, or, to each side of the equation. 5252 2 25 4 5252 2 x + 200 4 + 25 4 = Write x 2 + 5x + as a square. 5252 2 Rewrite 50 as a fraction with denominator 4. 5252 2 x + = 225 4 10-5

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Completing the Square (continued) Step 2: Solve the equation. Find the square root of each side. 5252 x + = 225 4 ± 5252 x + = 15 2 ± Simplify. 5252 x + = 15 2 or 5252 x += 15 2 – Write as two equations. x = 5or x = –10Solve for x. 10-5 Quick Check

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Completing the Square Solve x 2 + 10x – 16 = 0 by completing the square. Round to the nearest hundredth. Step 1: Rewrite the equation in the form x 2 + bx = c and complete the square. x 2 + 10x – 16 = 0 x 2 + 10x = 16Add 16 to each side of the equation. (x + 5) 2 = 41Write x 2 + 10x +25 as a square. x 2 + 10x + 25 = 16 + 25Add, or 25, to each side of the equation. 10 2 2 10-5

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Completing the Square (continued) Step 2: Solve the equation. x + 5 = ± 41Find the square root of each side. Use a calculator to find 41 x + 5± 6.40 x + 56.40orx + 5–6.40Write as two equations. Subtract 5 from each side.x6.40 – 5 or x –6.40 – 5 x1.40orx –11.40Simplify 10-5 Quick Check

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Completing the Square ALGEBRA 1 LESSON 10-5 Suppose you wish to section off a soccer field as shown in the diagram to run a variety of practice drills. If the area of the field is 6000 yd 2, what is the value of x? Define: width = x + 10 + 10 = x + 20 length = x + x + 10 + 10 = 2x + 20 Relate: length width = area Write: (2x + 20)(x + 20) = 6000 2x 2 + 60x + 400 = 6000 Step 1: Rewrite the equation in the form x 2 + bx = c. 2x 2 + 60x + 400 = 6000 2x 2 + 60x = 5600Subtract 400 from each side. x 2 + 30x = 2800Divide each side by 2. 10-5

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Completing the Square (continued) Step 2: Complete the square. x 2 + 30x + 255 = 2800 + 225 Add, or 225, to each side. 30 2 2 (x + 15) 2 = 3025Write x 2 + 30x + 255 as a square. Step 3: Solve each equation. (x + 15) = ± 3025 Take the square root of each side. x + 15 = ± 55Use a calculator. x + 15 = 55orx + 15 = –55 x = 40orx = –70Use the positive answer for this problem. The value of x is 40 yd. 10-5 Quick Check

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Completing the Square 1.x 2 + 14x = –43 2.3x 2 + 6x – 24 = 0 3.4x 2 + 16x + 8 = 40 Solve each equation by completing the square. If necessary, round to the nearest hundredth. –9.45, –4.55 –4, 2 –5.46, 1.46 10-5

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