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Thinking Mathematically Solving Quadratic Equations

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Using the FOIL Method to Multiply Binomials (ax + b)(cx +d) = axcx + axd + bcx + bd For example, multiply: (x + 3)(x + 4). F: First terms = xx = x 2 O: Outside terms = x4 = 4x I: Inside terms = 3x = 3x L: Last terms = 3 4 = 12 (x + 3)(x + 4) = x 2 + 4x + 3x + 12 = x 2 + 7x + 12

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A Strategy for Factoring x 2 + bx + c 1.Enter x as the first term of each factor. (x )(x ) = x 2 + bx + c 2.List all pairs of factors of the constant c. 3.Try various combinations of these factors. Select the combination in which the sum of the Outside and Inside products is equal to bx. (x + _ )(x + _ ) = x 2 + bx + c. 4.Check your work by multiplying the factors using the FOIL method. You should obtain the original trinomial.

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Solving Quadratic Equations by Factoring Standard Form of a Quadratic Equation ax 2 + bx + c = 0, a does not equal zero. The Zero-Product Principle If the product of two factors is 0, then one (or both) of the factors must have a value of 0. If AB = 0, then A = 0 or B = 0.

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Example Solving a Quadratic Equation Using the Zero-Product Principle Solve (x - 5)(x - 2) = 0 Set each factor equal to 0 and solve each equation x - 5 = 0 or x - 2 = 0 x = 5, x = 2 Check (5-5)(5-2) =?0(2-5)(2-2) =?0 0(3) = 0 true-3(0) = 0 true

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Solving a Quadratic Equation by Factoring 1.If necessary, rewrite the equation in the form ax 2 + bx + c = 0, moving all terms to one side, thereby obtaining zero on the other side. 2.Factor. 3.Apply the zero-product principle, setting each factor equal to zero. 4.Solve the equations in step 3. 5.Check the solutions in the original equation.

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Example using Factoring to Solve a Quadratic Equation Solve: x 2 - 2x = 35 1. Rewrite in standard form: x 2 - 2x - 35 = 0 2.Factor: (x - 7)(x + 5) = 0 3.Apply the zero-product principle: x - 7 = 0 or x + 5 = 0 4.Solve each equation: x = 7, x = -5 5.The original equation is satisfied by both numbers, so the solution set is {7, -5}.

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The Quadratic Formula The solutions of a quadratic equation in standard form ax 2 + bx + c = 0 with a not equal to zero, are given by the quadratic formula.

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Thinking Mathematically Solving Quadratic Equations

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