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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.StepbyStep:
1
Consider the x^{2} and x terms only.
We can complete the square on these terms, replacing them with a squared binomial minus a number.
We can complete the square on these terms, replacing them with a squared binomial minus a number.
2
Replace the x^{2} 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.
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).
Don't forget: The answer from Step 5 (2) comes from dividing the coefficient of x (4) by 2.
2^{2} = 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
Answer:
We have completed the square on the quadratic equation: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: x^{2} + 4x + 2^{2}.
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 (2^{2}).
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