


Solve by Factoring LessonsSeveral previous lessons explain the techniques used to factor expressions. This lesson focuses on an imporatant application of those techniques ??? solving equations. Why solve by factoring?The most fundamental tools for solving equations are addition, subtraction, multiplication, and division. These methods work well for equations like x + 2 = 10  2x and 2(x  4) = 0. But what about equations where the variable carries an exponent, like x^{2} + 3x = 8x  6? This is where factoring comes in. We will use this equation in the first example. The Solve by Factoring process will require four major steps:
First Examplex^{2} + 3x = 8x  6 Step 1The first step is to move all terms to the left using addition and subtraction. First, we will subtract 8x from each side. x^{2} + 3x  8x = 8x  8x  6 Now, we will add 6 to each side. x^{2}  5x + 6 = 6 + 6 With all terms on the left side, we proceed to Step 2. Step 2We identify the left as a trinomial, and factor it accordingly: (x  2)(x  3) = 0 We now have two factors, (x  2) and (x  3). Step 3We now set each factor equal to zero. The result is two subproblems: x  2 = 0 and x  3 = 0 Solving the first subproblem, x  2 = 0, gives x = 2. Solving the second subproblem, x  3 = 0, gives x = 3. Step 4The final step is to combine the two previous solutions, x = 2 and x = 3, into one solution for the original problem. x^{2} + 3x = 8x  6 Proceed to the next page for an explanation of the theory behind our method, and another example. 

