A48 APPENDIX Review A quadratic equation written in the form + + = ax bx c 0 2 is said to be in standard form . Sometimes a quadratic equation is called a second-degree equation because, when it is in standard form, the left side is a polynomial of degree 2. We discuss four algebraic ways of solving quadratic equations: by factoring, by using the square root method, by completing the square, and by using the quadratic formula. 3 Solve Quadratic Equations by Factoring When a quadratic equation is written in standard form, + + = ax bx c 0, 2 it may be possible to factor the expression on the left side as the product of two first-degree polynomials. Then, by using the Zero-Product Property and setting each factor equal to 0, we can solve the resulting linear equations to obtain the exact solutions of the quadratic equation. This approach leads us to a basic premise in mathematics. Whenever a problem is encountered, use techniques that reduce the problem to one you already know how to solve. In this instance, we are reducing quadratic equations to linear equations using the technique of factoring. DEFINITION Quadratic Equation A quadratic equation is an equation that is equivalent to one of the form + + = ax bx c 0 2 (1) where a, b, and c are real numbers and ≠ a 0. Need to Review? Factoring polynomials is discussed in Section A.3, pp. A2 7–A28. Solving a Quadratic Equation by Factoring Solve the equation: = + x x 2 3 2 Solution EXAMPLE 4 Put the equation = + x x 2 3 2 in standard form by subtracting x and 3 from both sides. = + − − = x x x x 2 3 2 3 0 2 2 Subtract x and 3 from both sides. The left side may now be factored as ( )( ) − + = x x 2 3 1 0 Factor. so that − = + = = = − x x x x 2 3 0 or 1 0 3 2 1 Use the Zero-Product Property. Solve. The solution set is { } −1, 3 2 . When the left side factors into two linear equations with the same solution, the quadratic equation is said to have a repeated solution . This solution is also called a root of multiplicity 2 , or a double root . Solving a Quadratic Equation by Factoring Solve the equation: − + = x x 9 6 1 0 2 Solution EXAMPLE 5 This equation is already in standard form, and the left side can be factored. ( )( ) − + = − − = x x x x 9 6 1 0 3 1 3 1 0 2 Factor.
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