Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top


Why is it that this must be true $q^2-p^3<0$ for x to have 3 distinct real roots?

share|cite|improve this question
Do you mean $q^2−p^3<0$? Square roots are always positive. – lhf Sep 17 '12 at 20:56
Yes, sorry- will adjust. – Alyosha Sep 19 '12 at 18:59
up vote 1 down vote accepted

This is the casus irreducibilis of the cubic. You need to use complex numbers even if the roots are all real. The proof needs Galois theory.

share|cite|improve this answer
Thank you. Before I try to delve into it, is learning Galois theory quickly feasible? – Alyosha Sep 20 '12 at 18:51
@Alyosha, not really, sorry. – lhf Sep 20 '12 at 20:44
Oh well. I'll return to this problem in a year or so. – Alyosha Sep 21 '12 at 17:07
@Alyosha, see… – lhf Sep 21 '12 at 18:07
Sorry to whip a dead post, but where I'm reading it says that you can prove this by 'considering the nature of stationary points'. Is this correct? – Alyosha Sep 27 '12 at 19:47

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.