Completing the Square

Completing the square rewrites a quadratic so one side is a perfect square trinomial. It is the move behind the quadratic formula and the way to find a parabola's vertex.

The core step: to complete $x^2 + bx$ , add $\left(\dfrac{b}{2}\right)^2$ . That is the number that makes the trinomial factor as a perfect square.

The magic number

Take half of the coefficient of $x$ and square it. For $x^2 + 12x$ , half of $12$ is $6$ , and $6^2 = 36$ .

Adding $36$ makes $x^2 + 12x + 36 = (x + 6)^2$ , a perfect square. That $\left(\dfrac{b}{2}\right)^2$ is the number you always add.

Using it to solve

To solve, add the magic number to both sides so the left becomes a perfect square, then solve by taking the square root.

Keep the equation balanced — whatever you add on the left you add on the right too. Then finish with the square-root method.

The same move converts a quadratic to vertex form: $x^2 + 2x - 3$ becomes $(x + 1)^2 - 4$ , so its parabola has vertex $(-1, -4)$ , the lowest point shown below.

Worked examples

Example 1: finding the number

What number completes the square for $x^2 + 12x$ ?

Half the middle coefficient

\dfrac{12}{2} = 6

Square it

6^2 = 36

Answer: $36$

Example 2: writing the square

Complete the square: $x^2 + 12x + 36 =$ ?

It factors as a perfect square

(x + 6)^2

Answer: $(x + 6)^2$

Example 3: using it to solve an equation

Solve $x^2 + 8x - 9 = 0$ by completing the square.

Move the constant to the right

x^2 + 8x = 9

Add half of 8, squared, to both sides

x^2 + 8x + 16 = 25

The left is now a perfect square

(x + 4)^2 = 25

Take the square root of both sides

x + 4 = \pm 5

Answer: $x = 1$ or $x = -9$

Try one yourself

Common questions

What number do I add?

Half of the x-coefficient, squared: $\left(\dfrac{b}{2}\right)^2$ . For $x^2 + 12x$ that is $6^2 = 36$ .

What if there's a coefficient on x-squared?

Factor it out of the x-squared and x terms first, complete the square inside, then account for the factor. The magic-number step applies to the bracket.

Why is completing the square useful?

It solves any quadratic, derives the quadratic formula, and converts to vertex form to read a parabola's vertex directly.

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Completing the Square

The magic number

Using it to solve

Worked examples

Example 1: finding the number

Example 2: writing the square

Example 3: using it to solve an equation

Try one yourself

Common questions

Keep learning

Solving Quadratics by Square Roots

The Quadratic Formula over the Complex Numbers

Graphing Quadratic Functions