# Derivation of Quadratic Formula

A Quadratic Equation looks like this:

And it can be solved using the Quadratic Formula:

That formula looks like magic, but you can follow the steps to see how it comes about.

## 1. Complete the Square

ax^{2} + bx + c has "x" in it twice, which is hard to solve.

But there is a way to rearrange it so that "x" only appears once. It is called Completing the Square (please read that first!).

Our aim is to get something like * x^{2} + 2dx + d^{2}*, which can then be simplified to

**(x+d)**^{2}So, let's go:

**a**

**c/a**on other side

**(b/2a)**to both sides

^{2}* The left hand side is now in the x^{2} + 2dx + d^{2} format, where "d" is "b/2a". So we can re-write it this way:*

Now x only appears once and we are making progress.

## 2. Now Solve For "x"

We will try to rearrange the equation to have just "x" on the left:

**b/2a**to right

**x**is on its own!

but let's simplify

**2a/2a**

Which is the Quadratic formula we all know and love: