1Cademy - Solving $$n^2 = 3n + 11$$ by Completing the Square

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Completing the Square

Example

Solving $n^2 = 3n + 11$ by Completing the Square

Solve $n^2 = 3n + 11$ by completing the square, demonstrating the procedure when variable terms must first be collected on one side and the linear coefficient is an odd number.

Step 1 — Collect the variable terms on one side. Subtract $3n$ from both sides to move all variable terms to the left:

$n^2 - 3n = 11$

Step 2 — Find $\left(\frac{1}{2} \cdot b\right)^2$ and add it to both sides. The coefficient of $n$ is $-3$ , so $b = -3$ . Compute: $\left(\frac{1}{2}(-3)\right)^2 = \left(-\frac{3}{2}\right)^2 = \frac{9}{4}$ . Add $\frac{9}{4}$ to both sides:

$n^2 - 3n + \frac{9}{4} = 11 + \frac{9}{4}$

Step 3 — Factor the perfect square trinomial. The left side factors as a binomial square using the subtraction form:

$\left(n - \frac{3}{2}\right)^2 = \frac{44}{4} + \frac{9}{4} = \frac{53}{4}$

Step 4 — Apply the Square Root Property:

$n - \frac{3}{2} = \pm\sqrt{\frac{53}{4}}$

Step 5 — Simplify the radical and solve. Apply the Quotient Property to the radical: $\sqrt{\frac{53}{4}} = \frac{\sqrt{53}}{\sqrt{4}} = \frac{\sqrt{53}}{2}$ . Since $53$ is prime, $\sqrt{53}$ cannot be simplified further. Add $\frac{3}{2}$ to both sides:

$n = \frac{3}{2} \pm \frac{\sqrt{53}}{2}$

Write as two solutions:

$n = \frac{3}{2} + \frac{\sqrt{53}}{2} \quad \text{or} \quad n = \frac{3}{2} - \frac{\sqrt{53}}{2}$

The solutions are $n = \frac{3 + \sqrt{53}}{2}$ and $n = \frac{3 - \sqrt{53}}{2}$ . This example combines two complications: first, the variable terms appear on both sides of the original equation, so they must be collected to one side before the procedure can begin. Second, the linear coefficient $-3$ is odd, so halving it produces the fraction $-\frac{3}{2}$ , and the resulting constant $\frac{9}{4}$ introduces fractions throughout the remaining steps. The radicand $53$ is not a perfect square, so the final solutions remain in radical form.

Updated 2026-04-21

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OpenStax Elementary Algebra

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