Solve by Factoring: Why does it work?

Examine the equation below:

ab = 0

If you let a = 3, then logivally b must equal 0. Similarly, if you let b = 10, then a must equal 0.

Now try letting a be some other non-zero number. You should observe that as long as a does not equal 0, b must be equal to zero.

To state the observation more generally, "If ab = 0, then either a = 0 or b = 0." This is an important property of zero which we exploit when solving by factoring.

When the example was factored into (x - 2)(x - 3) = 0, this property was applied to determine that either (x - 2) must equal zero, or (x - 3) must equal zero. Therefore, we were able to create two equations and determine two solutions from this observation.

A Second Example

5x³ = 45x

Step 1

Move all terms to the left side of the equation. We do this by subtracting 45x from each side.

5x³ - 45x = 45x - 45x
5x³ - 45x = 0.

Step 2

The next step is to factor the left side completely. We first note that the two terms on the left have a greatest common factor of 5x.

5x(x² - 9) = 0

Now, (x² - 9) can be factored as a difference between two squares.

5x(x + 3)(x - 3) = 0

We are left with three factors: 5x, (x + 3), and (x - 3). As explained in the "Why does it work?" section, at least one of the three factors must be equal to zero.

Step 3

Create three subproblems by setting each factor equal to zero.

1.   5x = 0
2.   x + 3 = 0
3.   x - 3 = 0

Solving the first equation gives x = 0. Solving the second equation gives x = -3. And solving the third equation gives x= 3.

Step 4

The final solution is formed from the solutions to the three subproblems.

x = -3, 0, 3

Visit the next page for another example.