Algebra/Completing the Square

Derivation
The purpose of "completing the square" is to either factor a prime quadratic equation or to more easily graph a parabola. The procedure to follow is as follows for a quadratic equation $$ y = ax^2+bx+c $$:

1. Divide everything by a, so that the number in front of $$ x^2 $$ is a perfect square (1):
 * $$ \frac{y}{a} = x^2 + \frac{b}{a}x + \frac{c}{a} $$

2. Now we want to focus on the term in front of the x. Add the quantity $$ \left(\frac{b}{2a}\right)^2 $$ to both sides:
 * $$ \frac{y}{a} + \left(\frac{b}{2a}\right)^2 = x^2 + \frac{b}{a}x +\left(\frac{b}{2a}\right)^2 + \frac{c}{a} $$

3. Now notice that on the right, the first three terms factor into a perfect square:
 * $$ x^2 + \frac{b}{a}x +\left(\frac{b}{2a}\right)^2 = \left(x + \frac{b}{2a}\right)^2 $$

Multiply this back out to convince yourself that this works.

4. Therefore the completed square form of the quadratic is:
 * $$ \frac{y}{a} + \left(\frac{b}{2a}\right)^2 = \left(x + \frac{b}{2a}\right)^2 + \frac{c}{a} $$ or, multiplying through by a,

$$ y = a\left(x+\frac{b}{2a}\right)^2 + c - \frac{b^2}{4a} $$

Explanation of Derivation


1. Divide everything by a, so that the number in front of $$ x^2 $$ is a perfect square (1):
 * $$ x^2 + \frac{b}{a}x + \frac{c}{a} = {a}$$

Think of this as expressing your final result in terms of 1 square x. If your initial equation is

2. Now we want to focus on the term in front of the x. Add the quantity $$ \left(\frac{b}{2a}\right)^2 $$ to both sides:
 * $$ \frac{y}{a} + \left(\frac{b}{2a}\right)^2 = x^2 + \frac{b}{a}x +\left(\frac{b}{2a}\right)^2 + \frac{c}{a} $$

3. Now notice that on the right, the first three terms factor into a perfect square:
 * $$ x^2 + \frac{b}{a}x +\left(\frac{b}{2a}\right)^2 = \left(x + \frac{b}{2a}\right)^2 $$

Multiply this back out to convince yourself that this works.

4. Therefore the completed square form of the quadratic is:
 * $$ \frac{y}{a} + \left(\frac{b}{2a}\right)^2 = \left(x + \frac{b}{2a}\right)^2 + \frac{c}{a} $$ or, multiplying through by a,

Example
The best way to learn to complete a square is through an example. Suppose you want to solve the following equation for x.