The quadratic formula is learned fairly quickly after factoring is taught in algebra but many people don't know why this strange beast even exists and where it came from ... x= [-b+-(b^2-4ac)^(1/2)]/2a (my square root is written as a one half power to those who think it looks like something is missing).
TRUE STORY: When I was first introduced to the quad formula I could never remember it. So I would rederive it every time if I wasn't sure on the formula. Here is how it's done.
ax^2+bx+c=0 (Standard form of an equation ... OK, fair enough.)
ax^2+bx=-c (There's my negative c at least ... now let's complete the square!)
x^2+(b/a)x=-c/a (I divided out the a so I could get x^2 with a coefficient of 1.
x^2+(b/a)x+(b/2a)^2=-c/a+(b/2a)^2 (I told you I was completing the square.)
(x+b/2a)^2=-c/a +(b/2a)^2 (Root both sides! Woohoo)
x+b/2a = +-(-c/a+(b^2/4a^2))^(1/2) (I'm just going to simplify now a bit)
x+b/2a = +-[(-4ac+b^2)/(4a^2)]^(1/2) (Apply the root to the denominator only.)
x+b/2a = +-(-4ac+b^2)^(1/2)/2a (Subtract b/2a on both sides to solve for x)
x= -b/2a+-(-4ac+b^2)^(1/2)/2a (Common Denominator. Agreed?)
x= [-b+-(-4ac+b^2)^(1/2)]/2a (Let us simply flip the terms in the root so that the positive one is first.)