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Completing the Square
(KS4, Year 10)

homequadratic equationscompleting the square
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:

complete the square

How to Complete the Square

Completing the square is easy.

Question

Complete the square on the quadratic equation shown below.
complete the square example

Step-by-Step:

1

Consider the x2 and x terms only. complete_the_square_step_1 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.

complete the square step 2

3

Write an x in the brackets.

complete the square step 3

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.

complete the square step 4

5

Look at the original equation. Find the number in front of the x term (called the coefficient). In our example, it is 4. complete_the_square_step_5 Divide the coefficient of x by 2.
4 ÷ 2 = 2

6

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

complete the square step 6

7

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

complete the square step 7 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.

complete the square step 8

9

Add the numbers outside of the brackets together.

complete the square step 9 1
− 4 + 6 = 6 − 4 = + 2
complete the square step 9 2

Answer:

We have completed the square on the quadratic equation: complete_the_square_answer

Lesson Slides

The slider below shows another real example of how to complete the square.

Completing the Square and Perfect Square Trinomials

Completing the square comes from perfect square trinomials. perfect_square_trinomial_mini 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: perfect_square_trinomial_rearranged_mini 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).
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This page was written by Stephen Clarke.

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