# 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, $x^2+2x+3$ 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

$x^{2}+10x+24 = 0$
We begin by moving the constant term to the right side of the equation.
$x^2 + 10x = -24$
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 $\blueD{25}$.
$x^2 + 10x \blueD{ + 25} = -24 \blueD{ + 25}$
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.
$x + 5 = \pm1$
Isolate $x$ to find the solution(s).
$x = -5\pm1$

### Example 2

$4x^{2}+20x+25 = 0$
First, divide the polynomial by $4$ (the coefficient of the $x^2$ term).
$x^2 + 5x + \dfrac{25}{4} = 0$
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 $\dfrac{5}{2}$, and squaring it gives us $\blueD{\dfrac{25}{4}}$, our constant term.
Thus, we can rewrite the left side of the equation as a squared term.
$( x + \dfrac{5}{2} )^2 = 0$
Take the square root of both sides.
$x + \dfrac{5}{2} = 0$
Isolate $x$ to find the solution.
The solution is: $x = -\dfrac{5}{2}$

## Practice

Problem 1
Complete the square to rewrite this expression in the form $(x + a)^2 + b$.
$x^{2}-2x+17$

Want more practice? Check out these exercises: