# Completing the square review

CCSS Math: HSA.REI.B.4, HSA.REI.B.4a, HSA.REI.B.4b, HSA.SSE.B.3, HSA.SSE.B.3b, HSF.IF.C.8, HSF.IF.C.8a

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

We're given a quadratic and asked to complete the square.

We begin by moving the constant term to the right side of the equation.

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 $\blueD{25}$.

We can now rewrite the left side of the equation as a squared term.

Take the square root of both sides.

Isolate $x$ to find the solution(s).

*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.

First, divide the polynomial by $4$ (the coefficient of the $x^2$ term).

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.

Take the square root of both sides.

Isolate $x$ to find the solution.

The solution is: $x = -\dfrac{5}{2}$

## Practice

*Want more practice? Check out these exercises:*