Derive the quadratic equation.

So in order to derive the quadratic equation we need to start with a quadratic that can represent any quadratic equation. Lets start with ax²+bx+c=0.

We will use the completing the square method to solve this. First lets make sure the x² term is all by itself with no coefficient by dividing through a so x²+(b/a)x+(c/a)=0.

Now lets complete the square: (x+(b/2a))²-(b/2a)²+(c/a)=0.

This looks pretty messy so lets rearrange so the x terms are on one side and expand the (b/2a)² term: 

(x+(b/2a))²=(b²/4a²)-(c/a).

Now we can make the right hand side one big fraction by multiplying c by 4a so that the denominators are the same:

(x+(b/2a))²=(b²-4ac)/4a². 

We can see the b²-4ac term already which is good. Now lets square root both sides so we can single out x. After that it is a simple matter of rearranging again (I can't write a square root or plus/minus sign here so I have skipped some steps).