What is the Discriminant Test in Math? Used to verify best factorization

I remember factoring equations in school. But we never got this fancy Discriminant Test the guy mentions in this question (one of the answers):

http://answers.yahoo.com/question/index;…

We never knew if we were finished; we just had to keep trying until our brains wore out.

Can one of you explain how to use this formula to me?

for your reference:

quadratic formula
ax^2 + bx + c = 0
x = [-b±√(b^2-4ac)] / 2a
discriminant => b^2-4ac
b^2-4ac > 0 => 2 roots
b^2-4ac = 0 => 1 root
b^2-4ac < 0 => no root. (imagine roots)