The First Algebraic Identity of Quadratic Expressions

In algebra, three identities are essential for factorizing quadratic expressions. These are known as the first algebraic identity, the second algebraic identity, and the third algebraic identity of quadratic expressions. Let’s look at the first algebraic identity of quadratic expressions.

The algebraic identities help you multiply the terms in 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 in particular, but let’s now look at the first algebraic identity of quadratic expressions.

The first 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 from 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, I will rewrite the left-hand expression 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 should use the first algebraic identity to turn an expression into a multiplication problem.

You can actually use the first algebraic identity to factorize quadratic expressions.