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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 (x+a)2+b.
For example, x2+2x+3 can be rewritten as (x+1)2+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.
x2+10x+24=0
We begin by moving the constant term to the right side of the equation.
x2+10x=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 25.
x2+10x+25=24+25
We can now rewrite the left side of the equation as a squared term.
(x+5)2=1
Take the square root of both sides.
x+5=±1
Isolate x to find the solution(s).
x=5±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.
4x2+20x+25=0
First, divide the polynomial by 4 (the coefficient of the x2 term).
x2+5x+254=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 52, and squaring it gives us 254, our constant term.
Thus, we can rewrite the left side of the equation as a squared term.
(x+52)2=0
Take the square root of both sides.
x+52=0
Isolate x to find the solution.
The solution is: x=52

Practice

Problem 1
Complete the square to rewrite this expression in the form (x+a)2+b.
x22x+17

Want more practice? Check out these exercises:

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