1Cademy - Completing the Square for $$m^2

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Completing the Square

Example

Completing the Square for $m^2 - 26m$

Complete the square for the expression $m^2 - 26m$ to form a perfect square trinomial, then express the result as a binomial square.

Step 1 — Identify $b$ . The coefficient of $m$ is $-26$ , so $b = -26$ .

Step 2 — Find $\left(\frac{1}{2}b\right)^2$ . Compute half of $-26$ : $\frac{1}{2} \cdot (-26) = -13$ . Square the result: $(-13)^2 = 169$ .

Step 3 — Add $169$ to the expression.

$m^2 - 26m + 169$

Rewrite as a binomial square. Because the linear term is negative, the factored form uses subtraction:

$m^2 - 26m + 169 = (m - 13)^2$

This example demonstrates how the completing-the-square procedure handles a negative linear coefficient. Halving $-26$ gives $-13$ , and squaring a negative number still produces a positive constant ( $169$ ). The negative sign in the original expression determines that the binomial square uses the subtraction form of the Binomial Squares Pattern: $(m - 13)^2$ .

Updated 2026-04-21

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OpenStax Elementary Algebra

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