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Completing the Square

(KS4, Year 10)

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. 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

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.