Solve by Factoring

Several 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 x² + 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:

1. Move all terms to one side of the equation, usually the left, using addition or subtraction.
2. Factor the equation completely.
3. Set each factor equal to zero, and solve.
4. List each solution from Step 3 as a solution to the original equation.

First Example

x² + 3x = 8x - 6

Step 1

The first step is to move all terms to the left using addition and subtraction. First, we will subtract 8x from each side.

x² + 3x - 8x = 8x - 8x - 6
x² - 5x = -6

Now, we will add 6 to each side.

x² - 5x + 6 = -6 + 6
x² - 5x + 6 = 0.

With all terms on the left side, we proceed to Step 2.

Step 2

We 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 3

We 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 4

The final step is to combine the two previous solutions, x = 2 and x = 3, into one solution for the original problem.

x² + 3x = 8x - 6
x = 2, 3

Proceed to the next page for an explanation of the theory behind our method, and another example.