# Completing the Square

## The Basics

Completing the Square is a method commonly used to solve quadratic equations. Often times, a quadratic equation can be factored and solved easily. However, there are plenty of times when an equation is not factorable. By completing the square, a quadratic equation originally in standard form is rewritten into vertex form.

The equation
$ax^2+bx+c=0$
is converted into
$a(x-h)^2+k=0$
through the process of completing the square.

## Perfect Square Trinomial

A perfect square trinomial is a big fancy word for a simple concept. It is a quadratic equation that factors perfectly into two identical binomials.
In general,
$(x-a)(x-a)=x^2-2ax+a^2$
The quadratic on the right is a perfect square trinomial. It is the square of a binomial.
Take the example of $x^2+6x+9$. Using basic algebra, it can be factored into $(x+3)(x+3)$ or $(x+3)^2$
By completing the square, one of the components of the equation has to be a factored perfect square trinomial.

## Completing the Square

Let the quadratic equation be $ax^2+bx+c=0$

• Step 1: Move the constant over to the other side of the equality

$ax^2+bx=-c$

• Step 2: Factor out the coefficient of the squared term

$a(x^2+\tfrac{b}{a})=-c$