Your expression for the discriminant can be re-arranged into

.

For unequal roots (ie. two distinct roots) you want .

NB (an acronym for the latin phrasenota benewhich translates asnote well, that is,what follows is very important so pay close attention, chum): We don't give a fetid dingos kidney whether the roots are real or not, as long as they're distinct. So the requirement is , NOT .

Let's assume . Since a is real, then (IF the assumption is true) you can solvethisquadratic to find real values of a. Therefore (IF the assumption is true) the discriminant ofthisquadratic is greater than zero.

From here, I'll let you fill in the steps that lead to , an impossibility for real values of b and c.

Hence the assumption is false. So . Q.E.D.