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Published byFranklin Dutton Modified over 2 years ago

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xx x M ODELING C OMPLETING THE S QUARE 111 111 111 x2x2 xxx x x x Use algebra tiles to complete a perfect square trinomial. Model the expression x 2 + 6x. Arrange the x-tiles to form part of a square. To complete the square, add nine 1-tiles. You have completed the square. x 2 + 6x + 9 = (x + 3) 2

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S OLVING BY C OMPLETING THE S QUARE x 2 + bx + = x + ) ( b 2 () b 2 2 2 To complete the square of the expression x 2 + bx, add the square of half the coefficient of x.

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Completing the Square What term should you add to x 2 – 8x so that the result is a perfect square? S OLUTION The coefficient of x is –8, so you should add, or 16, to the expression. ) ( –8–8 2 2 x 2 – 8x + ) ( –8–8 2 2 = x 2 – 8x + 16 = (x – 4) 2

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Completing the Square S OLUTION Divide each side by 2. 2x 2 – x = 2 Factor 2x 2 – x – 2 = 0 2x 2 – x – 2 = 0 x 2 – x = 1 1 2 Write original equation. Add 2 to each side. 1 2 x 2 – x + = 1 + ) ( 1 4 2 – 1 16 Add =, or ( 1 2 – 1 2 ) 1 16 2 ) ( 1 4 2 – to each side.

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Completing the Square ) 1 4 2 – ( x = 17 16 17 4 1 4 x – = 1 2 x 2 – x + = 1 + ) ( 1 4 2 – 1 16 Add =, or ( 1 2 – 1 2 ) 1 16 2 ) ( 1 4 2 – to each side. Write left side as a fraction. Find the square root of each side. Add to each side. 1 4 x = 1 4 17 4 1 4 The solutions are + 1.28 and – 0.78. 1 4 – 17 4 4

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Completing the Square You can check the solutions on a graphing calculator. 1 4 The solutions are + 1.28 and – 0.78. 1 4 – 17 4 4 C HECK

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C HOOSING A S OLUTION M ETHOD Investigating the Quadratic Formula Perform the following steps on the general quadratic equation ax 2 + bx + c = 0 where a 0. ax 2 + bx = – c x 2 + + = bx a – c– c a a x 2 + + = + b 2a2a ) ( – c– c a 2 b 2a2a ) ( 2 b 2a2a ) ( 2 x + = + – c– c a b2b2 4a24a2 Use a common denominator to express the right side as a single fraction. b 2a2a ) ( 2 x + = – 4ac + b 2 4a24a2 Subtract c from each side. Divide each side by a. Add the square of half the coefficient of x to each side. Write the left side as a perfect square.

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C HOOSING A S OLUTION M ETHOD b 2a2a b 2 4ac 2a2a x + = Use a common denominator to express the right side as a single fraction. Investigating the Quadratic Formula Use a common denominator to express the right side as a single fraction. b 2a2a ) ( 2 x + = – 4ac + b 2 4a 24a 2 x = – b 2a2a 2a2a b 2 4ac x =x = 2a2a –b–b b 2 4ac Find the square root of each side. Include ± on the right side. Solve for x by subtracting the same term from each side.

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