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Solving Quadratic Equations by Completing the Square

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Perfect Square Trinomials l Examples l x 2 + 6x + 9 l x x + 25 l x x + 36

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Rule for Completing the Square This is now a PST! So, it factors into this!

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Creating a Perfect Square Trinomial l In the following perfect square trinomial, the constant term is missing. X x + ____ l Find the constant term by squaring half the coefficient of the linear term. l (14/2) 2 X x + 49

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Perfect Square Trinomials l Create perfect square trinomials. l x x + ___ l x 2 - 4x + ___ l x 2 + 5x + ___ /4

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Solving Quadratic Equations by Completing the Square Solve the following equation by completing the square: Step 1: Move quadratic term, and linear term to left side of the equation

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Solving Quadratic Equations by Completing the Square Step 2: Find the term that completes the square on the left side of the equation. Add that term to both sides.

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Solving Quadratic Equations by Completing the Square Step 3: Factor the perfect square trinomial on the left side of the equation. Simplify the right side of the equation.

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Solving Quadratic Equations by Completing the Square Step 4: Take the square root of each side

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Solving Quadratic Equations by Completing the Square Step 5: Set up the two possibilities and solve

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Completing the Square-Example #2 Solve the following equation by completing the square: Step 1: Move quadratic term, and linear term to left side of the equation, the constant to the right side of the equation.

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Solving Quadratic Equations by Completing the Square Step 2: Find the term that completes the square on the left side of the equation. Add that term to both sides. The quadratic coefficient must be equal to 1 before you complete the square, so you must divide all terms by the quadratic coefficient first.

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Solving Quadratic Equations by Completing the Square Step 3: Factor the perfect square trinomial on the left side of the equation. Simplify the right side of the equation.

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Solving Quadratic Equations by Completing the Square Step 4: Take the square root of each side

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Example: Solve by completing the square. x 2 +6x-8=0 x 2 +6x-8=0 x 2 +6x=8 x 2 +6x+___=8+___ x 2 +6x+9=8+9 (x+3) 2 =17 Dont forget Dont forget: Whatever you add to one side of an equation, you MUST add to the other side!

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More Examples! 5x 2 -10x+30=0 x 2 -2x+6=0 x 2 -2x=-6 x 2 -2x+__=-6+__ x 2 -2x+1=-6+1 (x-1) 2 =-5 3x 2 -12x+18=0 x 2 -4x+6=0 x 2 -4x=-6 x 2 -4x+__=-6+__ x 2 -4x+4=-6+4 (x-2) 2 =-2

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Last Example! Write the quadratic function y=x 2 +6x+16 in vertex form. What is the vertex of the functions graph? y=x 2 +6x+16 y-16=x 2 +6x y-16+__=x 2 +6x+__ y-16+9=x 2 +6x+9 y-7=(x+3) 2 y=(x+3) 2 +7 If the equation, in vertex form, is y=(x+3) 2 +7, then the vertex must be (-3,7).

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Solving Quadratic Equations by Completing the Square Try the following examples. Do your work on your paper and then check your answers.

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