How to Prove the (a − b)² Formula Algebraically Step by Step

Square of a Binomial in Product Form

The square of the binomial $a-b$ is the product obtained when the binomial is multiplied by itself. Therefore, the expression $(a – b)^2$ can be written in product form as follows.

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

Expand (a − b)² using the Algebraic Properties

The multiplication of binomials can be performed by distributing the first binomial $a − b$ to each term of the second expression using the distributive property.

$=\,\,$ $(a-b) \times a$ $-$ $(a-b) \times b$

The order of the factors in each term of the expression can be changed using the commutative property.

$=\,\,$ $a \times (a-b)$ $-$ $b \times (a-b)$

Now, apply the distributive property one more time to distribute the coefficient of each square of a difference binomial $a – b$ to each term of the binomial.

$=\,\,$ $a \times a$ $-$ $a \times b$ $-$ $b \times a$ $-$ $b \times (-b)$

Now, use the commutative property one more time to write the factors in each term in a consistent order.

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

Evaluate each term by multiplying its factors

The expansion of the $a−b$ whole square has four terms, and each term is a product of two factors. These factors can now be multiplied to simplify the expression.

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

Simplify the Expression to Derive the (a − b)² Formula

The expansion of $a – b$ squared has two like terms, and they can be combined to simplify the expression further.

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