Completing the Square
(KS4, Year 10)

Completing the square is a way of simplifying a quadratic equation. From perfect square trinomials, we have seen that a squared binomial expands to a quadratic equation:

If we have the first two terms in a quadratic equation (the x² and x terms), we can write it as a squared binomial minus a number:

Understanding Completing the Square Using Equations

Let's look at the patterns in the equations:

Halve the Number in Front of the x...

The number in the brackets (2) is half the number in front of the x (4).

...Then Square and Subtract It

The number being subtracted from the squared brackets (2) is half the number in front of the x (4) squared (2²).

Understanding Completing the Square Using Geometry

The area of the square plus the area of the rectangle below equals x² + 4x:

Split the rectangle in half. Instead of one rectangle with an area of 4x, there are two rectangles each with an area of 2x:

Place the two rectangles by the side of the square:

We almost have a square. If we place a small square of area 2² in the space... we will complete the square:

Each side of this larger square is x + 2. Its area is (x + 2)². By adding up the areas we see that:

x2 + 2x + 2x + 22 = (x + 2)2	4 small areas = 1 big area
x2 + 2x + 2x + 22 = (x + 2)2	Add like terms
x2 + 4x + 22 = (x + 2)2

The original area was x² + 4x not x² + 4x + 2². We added 2² to complete the square. To balance the equation, we must subtract 2² from both sides:

x2 + 4x + 22 − 22 = (x + 2)2 − 22	Subtract 22 from both sides
x2 + 4x = (x + 2)2 − 22

Completing the Square in General

In general, we complete the square as: