Completing the Square

Completing the square rewrites a quadratic so one side becomes a perfect square trinomial — something that factors as $(x + p)^2$ . Once it is a perfect square, you can solve by taking the square root of both sides.

The heart of the method is one number: to complete $x^2 + bx$ , you add $\left(\dfrac{b}{2}\right)^2$ . That is the value that makes the trinomial fold up into a perfect square.

The number that completes the square

Take half of the coefficient of $x$ , then square it. For $x^2 + 6x$ , half of $6$ is $3$ , and $3^2 = 9$ .

Adding $9$ makes $x^2 + 6x + 9$ , which factors as $(x + 3)^2$ . That $\left(\dfrac{b}{2}\right)^2$ is the number you always add.

Solving by completing the square

Move the constant to the other side, add $\left(\dfrac{b}{2}\right)^2$ to both sides, and the left side becomes a perfect square. Then take the square root of both sides, remembering the plus-or-minus.

Keep the equation balanced — whatever you add on the left, add on the right too. Finish by solving the two resulting equations for $x$ .

Worked examples

Example 1: finding the number

What number completes the square for $x^2 + 8x$ ?

Half the middle coefficient

\dfrac{8}{2} = 4

Square it

4^2 = 16

Answer: $16$

Example 2: solving a quadratic

Solve $x^2 + 6x = 7$ by completing the square.

Add (b/2) squared to both sides

x^2 + 6x + 9 = 7 + 9

Factor the perfect square

(x + 3)^2 = 16

Take the square root

x + 3 = \pm 4

Solve both

x = 1 \text{ or } x = -7

Answer: $x = 1$ or $x = -7$

Example 3: a coefficient on x squared

Solve $2x^2 + 8x = 10$ by completing the square.

Divide every term by 2 so x squared stands alone

x^2 + 4x = 5

Add (b/2) squared to both sides

x^2 + 4x + 4 = 5 + 4

Factor the perfect square

(x + 2)^2 = 9

Take the square root

x + 2 = \pm 3

Solve both

x = 1 \text{ or } x = -5

Answer: $x = 1$ or $x = -5$

Try one yourself

Common questions

What number do I add to complete the square?

Half of the x-coefficient, squared: $\left(\dfrac{b}{2}\right)^2$ . For $x^2 + 6x$ that is $3^2 = 9$ .

Why do I add it to both sides?

To keep the equation balanced. Adding the number only to the left would change the equation; adding it to both sides preserves equality while making the left a perfect square.

What if there is a coefficient on x squared?

Divide every term by that coefficient first so the $x^2$ term stands alone, then complete the square on the result.

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Completing the Square

The number that completes the square

Solving by completing the square

Worked examples

Example 1: finding the number

Example 2: solving a quadratic

Example 3: a coefficient on x squared

Try one yourself

Common questions

Keep learning

Quadratic Formula

Factoring Trinomials (a = 1)

Introduction to Quadratic Functions