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).
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