The Second Algebraic Identity of Quadratic Expressions

In this entry you will look at the second algebraic identity of quadratic expressions. The algebraic identities help you 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 second algebraic identity for now.

The second 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 to the expression on the right-hand side, and the expression on the right-hand side to the expression on the left-hand side. But first let’s see why the two sides are equal:

$$\begin{array}{llll}\hfill {(a-b)}^2& =(a-b)(a-b)& \hfill & \\ \hfill & =a^2-ab-ba+b^2& \hfill & \\ \hfill & =a^2-2ab+b^2& \hfill & \end{array}$$

In the first example, you will rewrite the expression on the left so it becomes the expression on the right-hand side.

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 second algebraic identity to turn an expression that is a collection of terms into a multiplication problem. You can actually use the second algebraic identity to factorize quadratic expressions.