Completing the Square
(KS4, Year 10)

The Lesson

Completing the square is a way of simplifying a quadratic equation. Completing the square on a quadratic equation writes it as a squared binomial plus (or minus) a number:

How to Complete the Square

Completing the square is easy.

Question

Complete the square on the quadratic equation shown below.

Step-by-Step:

1

Consider the x2 and x terms only.

We can complete the square on these terms, replacing them with a squared binomial minus a number.

2

Replace the x2 and x terms with a squared bracket. Leave a gap inside the brackets for two terms. Leave a gap after the brackets for a number to be subtracted.

3

Write an x in the brackets.

4

Look at the original equation. Find the sign in front of the x term. In our example, it is +. Write this sign after the x in the brackets.

5

Look at the original equation. Find the number in front of the x term (called the coefficient). In our example, it is 4.

Divide the coefficient of x by 2.
4 ÷ 2 = 2

6

Write the answer (2) in the gap in the brackets.

7

Square the answer from Step 5 (2).
22 = 2 × 2 = 4
Write it in the gap after the sign.

Don't forget: The answer from Step 5 (2) comes from dividing the coefficient of x (4) by 2.

8

Consider the whole of the equation.

9

Add the numbers outside of the brackets together.

− 4 + 6 = 6 − 4 = + 2

Answer:

We have completed the square on the quadratic equation:

Lesson Slides

The slider below shows another real example of how to complete the square. Open the slider in a new tab

Completing the Square and Perfect Square Trinomials

Completing the square comes from perfect square trinomials.

A perfect square trinomial is the result of squaring a binomial.
  • A binomial is two terms added (or subtracted) together: x + 2.
  • A squared binomial means multiplying the binomial by itself: (x + 2)2.
    (x + 2)2 = (x + 2) × (x + 2)
  • A trinomial is three terms added (or subtracted) together: x2 + 4x + 22.
This can be rearranged:

Let's look at the patterns in the equations:
  • The number in the brackets (2) is half the number in front of the x (4).
  • The number being subtracted from the squared brackets (2) is half the number in front of the x (4) squared (22).
Help Us To Improve Mathematics Monster

  • Do you disagree with something on this page?
  • Did you spot a typo?
Please tell us using this form

See Also

What is a quadratic equation? What is a square number? What is a coefficient?