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:
Start with  
Divide the equation by a  
Put c/a on other side  
Add (b/2a)^{2} to both sides  ^{} 
The left hand side is now in the x^{2} + 2dx + d^{2} format, where "d" is "b/2a" So we can rewrite it this way: 

"Complete the Square"  ^{} 
Now x only appears once and we are making progress.
2. Now Solve For "x"
Now we just need to rearrange the equation to leave "x" on the left
Start with  ^{} 
Square root  
Move b/2a to right  
That is actually solved! But let's simplify it a bit: 

Multiply right by 2a/2a  
Simplify: 
Which is the Quadratic formula we all know and love: