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M ODELING C OMPLETING THE S QUARE

x. x. x. x. x. x. x 2. x. x. x. M ODELING C OMPLETING THE S QUARE. Use algebra tiles to complete a perfect square trinomial. Model the expression x 2 + 6 x. Arrange the x -tiles to form part of a square. 1. 1. 1. 1. 1. 1. To complete the square, add nine 1-tiles. 1. 1.

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M ODELING C OMPLETING THE S QUARE

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  1. x x x x x x x2 x x x MODELING COMPLETING THE SQUARE Use algebra tiles to complete a perfect square trinomial. Model the expression x2 + 6x. Arrange the x-tiles to form part of a square. 1 1 1 1 1 1 To complete the square, add nine 1-tiles. 1 1 1 x2 + 6x + 9 = (x + 3)2 You have completed the square.

  2. SOLVING BY COMPLETING THE SQUARE 2 2 ) ( ) ( b b x2 + bx + = x + 2 2 To complete the square of the expressionx2 + bx, add the square of half the coefficient ofx.

  3. Completing the Square 2 ( ) –8 The coefficient of x is –8, so you should add , or 16, to the expression. 2 2 ( ) –8 x2 – 8x + = x2 – 8x + 16 = (x – 4)2 2 What term should you add tox2 – 8xso that the result is a perfect square? SOLUTION

  4. Completing the Square 1 x2 – x = 1 2 2 2 ( 2 ) ( ) 1 1 1 1 ( ) 1 1 1 – – Add = , or x2–x+ = 1 + – • 2 16 2 2 4 16 4 to each side. Factor2x2 – x – 2 = 0 SOLUTION 2x2 – x – 2 = 0 Write original equation. 2x2 – x = 2 Add 2 to each side. Divide each side by 2.

  5. Completing the Square 2 ( ) 1 1 x2–x+ = 1 + – 16 1 4 2 2 ( ) 1 17 = x – 16 4 1 17 17 17 17 x – = ± 4 4 4 4 4 1 1 x = ± Add to each side. 4 2 2 4 ( ) ( ) 1 1 1 1 – – Add = , or • 2 16 2 4 to each side. 1 1 – The solutions are +  1.28 and  – 0.78. 4 4 Write left side as a fraction. Find the square root of each side.

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

  7. Perform the following steps on the general quadratic equation ax2 + bx + c = 0 where a 0. –c bx x2+ + = a a ( ) ( ) bx –c b b 2 2 x2+ + = + a a 2a 2a ( ) b –c b2 2 x + = + a 2a 4a2 ( ) –4ac + b2 b 2 x + = 4a2 2a CHOOSING A SOLUTION METHOD Investigating the Quadratic Formula ax2 + bx = –c Subtractcfrom each side. Divide each side bya. Add the square of half the coefficient ofxto each side. Write the left side as a perfect square. Use a common denominator to express the right side as a single fraction.

  8. ( ) b –4ac + b2 2 Use a common denominator to express the right side as a single fraction. x + = 2a 4a2 2 ± - b b 4 ac 2 ± - x + = b 4 ac 2a 2a 2a b x= – 2a 2 ± - b 4 ac –b x= 2a CHOOSING A SOLUTION METHOD Investigating the Quadratic Formula Find the square root of each side.Include±on the right side. Solve for x by subtracting the same term from each side. Use a common denominator to express the right side as a single fraction.

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