1Cademy - Completing the Square for $$p^2 + \frac{1}{2}p$$

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Completing the Square

Example

Completing the Square for $p^2 + \frac{1}{2}p$

Complete the square for the expression $p^2 + \frac{1}{2}p$ to form a perfect square trinomial, then express the result as a binomial square.

Step 1 — Identify $b$ . The coefficient of $p$ is $\frac{1}{2}$ , so $b = \frac{1}{2}$ .

Step 2 — Find $\left(\frac{1}{2}b\right)^2$ . Compute half of $\frac{1}{2}$ : $\frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}$ . Square the result: $\left(\frac{1}{4}\right)^2 = \frac{1}{16}$ .

Step 3 — Add $\frac{1}{16}$ to the expression.

$p^2 + \frac{1}{2}p + \frac{1}{16}$

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

$p^2 + \frac{1}{2}p + \frac{1}{16} = \left(p + \frac{1}{4}\right)^2$

This example demonstrates the completing-the-square procedure when the linear coefficient is itself a fraction rather than an integer. Taking half of $\frac{1}{2}$ gives $\frac{1}{4}$ , and squaring $\frac{1}{4}$ gives $\frac{1}{16}$ . Each step involves multiplying fractions — $\frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}$ and $\frac{1}{4} \cdot \frac{1}{4} = \frac{1}{16}$ — but the overall procedure remains the same three steps used for integer coefficients. The result is a perfect square trinomial whose constant term is a small fraction.

Updated 2026-04-21

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OpenStax Elementary Algebra

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