5.1.3 Solving Quadratic Equations by Graphing and Factoring Key Objectives • Solve quadratic equations by graphing or factoring. • Determine a quadratic function from its roots. Key Terms
• A zero of a function is any value of the input x such that the output equals zero, f ( x ) = 0. • The root of an equation is any value of the variable that makes the equation true. Example 1 Finding Zeros by Using a Graph or Table Unlike linear functions, which have no more than one zero, quadratic functions can have two zeros. These two zeros are always symmetric about the axis of symmetry. The zeros of a function can be identified from the function’s graph because the zeros are the x -intercepts.
Example 2 Finding Zeros by Factoring
A function’s zeros can also be found algebraically. To find a function’s zeros, set the related equation equal to 0 (i.e., let f ( x ) = 0) and solve the equation for x . The solutions to the related equation, or the roots of the equation, represent the zeros of the function. The roots of some quadratic equations can be found by factoring and appling the Zero Product Property, which states that if the product of two quantities equals 0, at least one of the two quantities equals 0. Zero Product Property For all real numbers a and b , if ab = 0, then a = 0 or b = 0.
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