# Difference between revisions of "Completing the Square"

## The Basics

Completing the Square is a method used to solve When a quadratic is hard to factor or not factorable at all, this method rewrites the quadratic equation originally in into which like a factored quadratic is much easier to solve.

The equation
$ax^2+bx+c=0$
is converted into
$a(x-h)^2+k=0$

and $x=\pm\sqrt{\frac{-k}{a}}+h$

## 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
In general,
$x^2-2ax+a^2=(x-a)(x-a)=(x-a)^2$
The quadratic on the left is a perfect square trinomial. It is the square of a binomial.

### Example of a Perfect Square Trinomial

Take the example of $x^2+6x+9$. It can be factored into $(x+3)(x+3)$ whcih is equal to $(x+3)^2$
This concept is important because in the process of completing the square, one of the components of the equation has to be a factored perfect square trinomial.

#### Relationship between terms

• Note that the coefficient of the middle term is twice the square root of the constant term in a quadratic equation.
• In other words, the constant term is the square of half of the coefficient of x.
• NOTE that this holds only true if the coefficient of $x^2$ is 1.

In general given the perfect trinomial and the equivalent squared binomial $x^2+bx+c=(x+d)^2$
Then $x^2+bx+c=x^2+2dx+d^2$
So,
$b=2d$ and $c=d^2$
We can rearrange equalities to obtain the following: $c=(\tfrac{b}{2})^2$

## Procedure for Completing the Square

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

• : Move the constant term c over to the other side of the equal sign.
• : Factor out a, the coefficient of the squared term.
• : Complete the quadratic in the parenthesis to make a perfect square trinomial. Remember that in general the constant of the perfect square trinomial is the square of half of the coefficient of x.
• : Add the equivalent added to the left side of the equation, to the right side of the equation to maintain the equality. Remember to multiply the obtained constant by a when added to the right side.
• : Factor the left side, simplify the right side and bring it over to the left.
• Step 6(optional): If asked for, solve for x.

<template>AlignEquals

Though the process of completing the square is finalized, it does not look like the

Let <template>AlignEquals

|e1l=-h
|e1r=\frac{b}{2a}
|e2l=k
|e2r=c-\frac{b^2}{4a}
</template>


We can now substitute these values into the equation in Step 5 and obtain
$a(x-h)^2+k=0$

### Example with a = 1

All the different letters are bound to get confusing. It may be easier to understand the process using an example with real numbers. Suppose we are given the quadratic $x^2-12x+5$, and asked to solve for x. The first thing we do is see if it can be factored: it can't. So to solve for x, we will complete the square.

### Example with a not 1

Given $3x^2+2x-7$