Completing the Square: A Walkthrough That Finally Clicks

Completing the square is one of those algebra moves that students see once and forget. But it is the single technique behind the quadratic formula, vertex form of a parabola, and several common calculus integrals. Once you internalise the trick, you have a tool you'll use forever.

The core idea

The squared binomial $(x + h)^2$ expands to $x^2 + 2hx + h^2$. To turn any expression $x^2 + bx$ into a perfect square, you need to add $\left(\frac{b}{2}\right)^2$. That is the entire trick.

Worked example: monic case

Complete the square on $x^2 + 6x + 5$.

1. Take half the linear coefficient: $b/2 = 3$.
2. Square it: $9$.
3. Rewrite: $x^2 + 6x + 9 - 9 + 5 = (x + 3)^2 - 4$.

We added 9 and subtracted 9 — net zero, but the first three terms now form a perfect square.

Worked example: non-monic case

Complete the square on $2x^2 + 12x + 7$.

1. Factor 2 out of the first two terms: $2(x^2 + 6x) + 7$.
2. Inside the bracket, complete the square: $x^2 + 6x + 9 - 9 = (x+3)^2 - 9$.
3. Substitute back: $2((x+3)^2 - 9) + 7 = 2(x+3)^2 - 18 + 7 = 2(x+3)^2 - 11$.

Application 1: solving quadratics

To solve $x^2 + 6x + 5 = 0$:
$(x + 3)^2 - 4 = 0 \Rightarrow (x+3)^2 = 4 \Rightarrow x + 3 = \pm 2 \Rightarrow x = -1, -5$.

Same answer as the quadratic formula, derived from scratch.

Application 2: vertex of a parabola

$y = 2x^2 + 12x + 7 = 2(x + 3)^2 - 11$ is in vertex form $y = a(x - h)^2 + k$. Vertex is at $(h, k) = (-3, -11)$, opening upward (since $a > 0$). You can read this off without calculus.

Application 3: integration

Integrals like $\int \frac{dx}{x^2 + 4x + 13}$ resist direct attack but yield to completing the square: $x^2 + 4x + 13 = (x + 2)^2 + 9$, then substitute $u = x + 2$ to recognize an arctangent.

Common mistakes

• Forgetting to subtract what you added — the expression must remain equal to itself.
• Not factoring out the leading coefficient in non-monic cases first.
• Halving the wrong coefficient — it's the linear coefficient $b$, not the leading $a$.

Try with the AI Quadratic Solver

The Quadratic Solver shows the completing-the-square approach side-by-side with the quadratic formula.

Related references:

• Factor Calculator — the alternative path to roots
• Equation Solver — broader equation-solving toolkit
• Integral Calculator — for the calculus application above