 # The Third Algebraic Identity of Quadratic Expressions

In addition to the first and second algebraic identities of quadratic expressions, the third algebraic identity is very important. In this entry you will learn about that third algebraic identity.

These algebraic identities help you to expand parentheses quickly, factorize some types of expressions, solve some types of equations and simplify some types of fractions. In other entries I will go through all the different areas, but let’s focus on the third algebraic identity for now.

Formula

 $\left(a-b\right)\left(a+b\right)={a}^{2}-{b}^{2}$

The third algebraic identity consists of a left-hand side expression, an equal sign and a right-hand side expression. This means that you can transform the expression on the left-hand side into the expression on the right-hand side, and from the expression on the right-hand side into the expression on the left-hand side. But first let’s see why the two sides are equal: $\begin{array}{llll}\hfill \left(a-b\right)\left(a+b\right)& ={a}^{2}+ab-ba-{b}^{2}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & ={a}^{2}-{b}^{2}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

In the first example, you will rewrite the left expression to make it into the expression on the right-hand side.

Example 1

Expand $\left(x-2\right)\left(x+2\right)$

 $\left(x-2\right)\left(x+2\right)={x}^{2}-4$

because $\begin{array}{llll}\hfill \left(x-2\right)\left(x+2\right)& ={x}^{2}+2x-2x-{2}^{2}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & ={x}^{2}-4\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

Keep note that the middle term disappears because the parentheses have different signs in front of the last term. But what happens when you go the opposite way— from the right-hand side of the formula to the left-hand side? You can use the third algebraic identity to turn an expression that’s a collection of terms into a multiplication problem. You can actually use the third algebraic identity to factorize quadratic expressions.

Example 2

Factorize ${x}^{2}-4$

 ${x}^{2}-4=\left(x+2\right)\left(x-2\right)$

because $\begin{array}{llll}\hfill {x}^{2}-4& =\phantom{\rule{-0.17em}{0ex}}\left(\sqrt{{x}^{2}}-\sqrt{4}\right)\phantom{\rule{-0.17em}{0ex}}\left(\sqrt{{x}^{2}}+\sqrt{4}\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\left(x-2\right)\left(x+2\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

The third algebraic identity appears all the time, so it should be your new best friend!

Example 3

This example is often used on tests and exams. Follow carefully when I factorize ${x}^{2}-1$:

 ${x}^{2}-1=\left(x+1\right)\left(x-1\right),$

since ${1}^{2}=1$. So don’t get tricked!