# Difference of Squares Factoring Guide

Factoring is a method that allows to see expressions in a different way or form. A factored form can allow us to simplify expressions which can help us in finding solutions to equations.

If we are given the form , we can express this as a factored form as follows:

If we were to verify this by going in the reverse way by expanding , the calculations would be like this:

Examples

Let’s demonstrate this factoring method through many examples.

Example 1

Factor .

Solution:

We have and . So and the positive part of the square root of 9 is 3 so . The factored form of is simply .

Example 2

Solve the equation .

Solution:

In this one, it wants you to solve the equation. In other words, find x-values such that . To do this, we factor first and solve for the x-values from the factors.

We have and . So and the positive part of the square root of 144 is 12 so . The factored form of is simply .

From , we have . Now we equate each factor to zero and solve for x.
In and in . Our solutions for are 12 and -12.

Example 3

Factor .

Solution:

Source: http://quicklatex.com/cache3/6f/ql_7d234bca80bcf63eaeb505bcce15ea6f_l3.png

This one does not look as obvious but if you know your exponent laws well, can be expressed as .

Here we have and . The factored form of is .

Notice that we can factor further by factoring to .

Example 4

Solve for in the equation .

Solution:

Source: http://quicklatex.com/cache3/11/ql_ba1b4bc21549df33d84bd17dd3e41711_l3.png

The solution for would be . Note that . Thus, the solution can also be expressed as . Remember also that for example can be expressed as .

Example 5 (“Complex” Case)

Solve for in the equation .

Solution:

We have

Source: http://quicklatex.com/cache3/57/ql_f25e56f564aa3fc91b0bdc9bf469a957_l3.png

Here we do not have a difference of squares situation. The expression cannot be factored. If we were to take the square root of , it would not exist in the real numbers.

However, if we use complex/imaginary numbers where then we can factor. Then we have . The imaginary numbered solutions to are .

Notes

• It is important to note that is not the same as . This is because order matters and factoring a (-1) from gives .
• If you are new to factoring difference of squares, practice is recommended. Know your square numbers such as 1, 4, 9 ,16 , 25, 36 ,49,64, 81, 100, 121, 144, and 169 really well.

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