1Cademy - Completing the Square

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Binomial Squares Pattern

Definition

Completing the Square

Completing the square is an algebraic technique for determining the constant that must be added to a two-term expression like $x^2 + 6x$ to transform it into a perfect square trinomial — one that can be factored as the square of a binomial. The technique is motivated by the need to apply the Square Root Property, which requires the variable side of an equation to be a perfect square. When the expression is not already a perfect square, completing the square provides a way to make it one.

The method works by reversing the Binomial Squares Pattern. Since $(a + b)^2 = a^2 + 2ab + b^2$ , the middle term $2ab$ is always twice the product of the binomial's two terms. Consider the expression $x^2 + 6x$ . The first term $x^2$ identifies $a = x$ . The linear term $6x$ corresponds to $2ab$ in the pattern, so $2 \cdot x \cdot b = 6x$ , which means $b = \frac{1}{2} \cdot 6 = 3$ . To find the missing constant:

Take half the coefficient of the linear term: $\frac{1}{2} \cdot 6 = 3$ . This gives the value of $b$ .
Square that result to obtain $b^2$ : $3^2 = 9$ .

Adding $9$ to the original expression produces a perfect square trinomial: $x^2 + 6x + 9 = (x + 3)^2$ . The sign between the two original terms determines which form of the Binomial Squares Pattern applies — a positive linear term uses $(a + b)^2$ , while a negative linear term uses $(a - b)^2$ .

Updated 2026-04-21

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