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# Solving Equations by Factoring

Definition: Any equation that can be put in the form ax2 + bx + c = 0 where a, b, and c are real numbers (a 0), is called a quadratic equation. The standard form for a quadratic equation:

ax2 + bx + c = 0

Zero-Factor Theorem: For real numbers A and B, if AÂ·B = 0 then A = 0 or B = 0, or both.

 WHAT TO DO: HOW TO DO IT: 1. Solve the equation x 2 − 5x − 6 = 0 Scratch: Find a numbers whose product is 6 with difference of 5 , larger sign â€œ−â€. Set each factor equal to zero: The solution: x 2 − 5x − 6 = 0 (x − 6)(x + 1) = 0 (x − 6) = 0 or (x + 1) = 0 x = 6 or x = − 1 Check the equation x 2 − 5x − 6 = 0 (6)2 − 5(6) − 6 = 0 , (-1)2 − 5(-1) − 6 = 0 2. Solve the equation x 2 = 7x − 12 Rewrite as a standard equation: Scratch: Find a pair of numbers whose product is 12 with sum of 7. Set each factor equal to zero: The solution: x 2 = 7x − 12 x 2 − 7x + 12 = 0 (x − 4)(x − 3) = 0 (x − 4) = 0 or (x − 3) = 0 x = 4 or x = 3 Check the equation x 2 = 7x − 12 (4)2 = 7(4) − 12 , (3)2 = 7(3) − 12 2. Solve the equation x 2 = 12x Rewrite as a standard equation: Set each factor equal to zero: The solution: x 2 = 12x x 2 − 12x = 0 xÂ·(x − 12) = 0 x = 0 or (x − 12) = 0 x = 0 or x = 12 Check the equation x 2 = 12x (0)2 = 12(0) , (12)2 = 12(12)