$(a-b)^2$ identity

Formula

$(a-b)^2 \,=\, a^2-2ab+b^2$

What is the (a−b)² Formula?

$a$ minus $b$, squared, equals $a$ squared minus two $a$ $b$ plus $b$ squared.

When $a$ and $b$ represent two variables, subtracting $b$ from $a$ gives their difference, written as $a-b$ in algebra. Squaring this difference gives the expression $(a-b)^2$, which represents the square of a difference in mathematics.

The square of $a−b$ can be read in two different ways, and you can use either one.

1. $a$ minus $b$ squared
2. $a$ minus $b$ whole square

Now, let us learn how to expand the $a$ minus $b$ squared (or $a$ minus $b$ whole square) expression and understand how to use it in mathematics with some simple and easy-to-understand examples.

How to Use the (a−b)² Formula

The $a$ minus $b$ squared formula is mainly used in two different cases in mathematics.

Expansion

We already know what the $a$ minus $b$ squared represents. Now, let us learn how to expand the algebraic expression $(a-b)^2$ in terms of the variables $a$ and $b$.

$\implies$ $(a-b)^2$ $\,=\,$ $a^2-2ab+b^2$

The $a$ minus $b$ whole square is expanded as $a$ squared minus two $a$ $b$ plus $b$ squared, and this expansion is used as a formula when an expression is already written as the square of a difference (a binomial).

The above example explains how a binomial in the form of the square of a difference is expanded step by step into a three-term expression using the $a-b$ whole square formula.

Factorization

So far, we have used the $a$ minus $b$ squared formula to expand expressions. The same formula can also be used in reverse to factor a trinomial into the square of a difference.

$\implies$ $a^2-2ab+b^2$ $\,=\,$ $(a-b)^2$

The thee-term expression can be factored using the $a$ minus $b$ whole square formula.

The above example demonstrates how to factor a trinomial whose first and last terms are perfect squares and whose middle term is $-2ab$ into the square of a difference using the $a$ minus $b$ whole square formula.

The above two simple examples show how to use the $a$ minus $b$ squared formula in mathematics.

Verification of the (a−b)² Formula

You have learned how to expand and factor expressions using the $a$ minus $b$ squared formula, but it’s natural to question whether it is really correct mathematically, and we can verify it using simple arithmetic with numbers.

These two numerical examples clearly prove that the $a$ minus $b$ whole square is equal to $a$ squared minus two $a$ $b$ plus $b$ squared. You can also verify the $a-b$ squared formula by substituting different numbers and observing that both sides are equal. For this reason, it is also called the $(a-b)^2$ identity.

Other Forms of the Square of a Difference

In mathematics, the square of a difference formula can be written using different variables such as $x$ and $y$ instead of $a$ and $b$. Therefore, $(a-b)^2$ and $(x-y)^2$ represent the same square of a difference formula, with only the variables changing.

You have learned what the $(a-b)^2$ formula is and how it is used in mathematics, including arithmetic verification to check its correctness. Next, we will learn how to derive the $(a-b)^2$ algebraic identity clearly and step by step in two different methods.