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

Learn Before

Completing the Square

Example

Completing the Square for $u^2 - 9u$

Complete the square for the expression $u^2 - 9u$ to form a perfect square trinomial, then express the result as a binomial square.

Step 1 — Identify $b$ . The coefficient of $u$ is $-9$ , so $b = -9$ .

Step 2 — Find $\left(\frac{1}{2}b\right)^2$ . Compute half of $-9$ : $\frac{1}{2} \cdot (-9) = -\frac{9}{2}$ . Square the result: $\left(-\frac{9}{2}\right)^2 = \frac{81}{4}$ .

Step 3 — Add $\frac{81}{4}$ to the expression.

$u^2 - 9u + \frac{81}{4}$

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

$u^2 - 9u + \frac{81}{4} = \left(u - \frac{9}{2}\right)^2$

This example demonstrates what happens when the linear coefficient is an odd integer. Unlike even coefficients (such as $14$ or $-26$ ), taking half of an odd number produces a fraction — here, $\frac{1}{2} \cdot (-9) = -\frac{9}{2}$ . Squaring that fraction gives another fraction: $\left(-\frac{9}{2}\right)^2 = \frac{81}{4}$ . As a result, both the constant added to complete the square and the number inside the binomial square are fractions rather than integers. The procedure itself is identical to the integer-coefficient cases — halving a negative odd number still yields a negative result, and squaring it still produces a positive constant.

Updated 2026-04-21

Contributors are:

Who are from:

References

OpenStax Elementary Algebra

Learn Before

Related

Learn After