The Lesson

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:

Lesson Slides

Completing the square works when the terms in the squared binomial are subtracted from each other. The slider below shows another real example of completing the square. Open the slider in a new tab