Completing the square review

Completing the square is a technique for factoring quadratics. This article reviews the technique with examples and even lets you practice the technique yourself.

What is completing the square?

Completing the square is a technique for rewriting quadratics in the form left parenthesis, x, plus, a, right parenthesis, start superscript, 2, end superscript, plus, b.
For example, x, start superscript, 2, end superscript, plus, 2, x, plus, 3 can be rewritten as left parenthesis, x, plus, 1, right parenthesis, start superscript, 2, end superscript, plus, 2. The two expressions are totally equivalent, but the second one is nicer to work with in some situations.

Example 1

We're given a quadratic and asked to complete the square.
x, start superscript, 2, end superscript, plus, 10, x, plus, 24, equals, 0
We begin by moving the constant term to the right side of the equation.
x, start superscript, 2, end superscript, plus, 10, x, equals, minus, 24
We complete the square by taking half of the coefficient of our x term, squaring it, and adding it to both sides of the equation. Since the coefficient of our x term is 10, half of it would be 5, and squaring it gives us start color blueD, 25, end color blueD.
x, start superscript, 2, end superscript, plus, 10, x, start color blueD, plus, 25, end color blueD, equals, minus, 24, start color blueD, plus, 25, end color blueD
We can now rewrite the left side of the equation as a squared term.
left parenthesis, x, plus, 5, right parenthesis, start superscript, 2, end superscript, equals, 1
Take the square root of both sides.
x, plus, 5, equals, plus minus, 1
Isolate x to find the solution(s).
x, equals, minus, 5, plus minus, 1
Want to learn more about completing the square? Check out this video.

Example 2

We're given a quadratic and asked to complete the square.
4, x, start superscript, 2, end superscript, plus, 20, x, plus, 25, equals, 0
First, divide the polynomial by 4 (the coefficient of the x, start superscript, 2, end superscript term).
x, start superscript, 2, end superscript, plus, 5, x, plus, start fraction, 25, divided by, 4, end fraction, equals, 0
Note that the left side of the equation is already a perfect square trinomial. The coefficient of our x term is 5, half of it is start fraction, 5, divided by, 2, end fraction, and squaring it gives us start color blueD, start fraction, 25, divided by, 4, end fraction, end color blueD, our constant term.
Thus, we can rewrite the left side of the equation as a squared term.
left parenthesis, x, plus, start fraction, 5, divided by, 2, end fraction, right parenthesis, start superscript, 2, end superscript, equals, 0
Take the square root of both sides.
x, plus, start fraction, 5, divided by, 2, end fraction, equals, 0
Isolate x to find the solution.
The solution is: x, equals, minus, start fraction, 5, divided by, 2, end fraction

Practice

Problem 1
Complete the square to rewrite this expression in the form left parenthesis, x, plus, a, right parenthesis, start superscript, 2, end superscript, plus, b.
x, start superscript, 2, end superscript, minus, 2, x, plus, 17

Want more practice? Check out these exercises: