Quadratic Formula

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.)
x= [-b+-(b^2-4ac)^(1/2)]/2a


Joshua C.


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