# Tartaglia cubic roots problem

$x=\sqrt[3]{q+\sqrt{q^2-p^3}}-\sqrt[3]{-q+\sqrt{q^2-p^3}}$

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

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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

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.

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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 math.stackexchange.com/questions/152818/… – 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
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