What Is Completing the Square?
Completing the Square
Completing the square is a way of simplifying 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
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:
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