Prove a relation between the coefficients of a depressed cubic.

Question

The equation $x^{3}+px^{2}+q=0$ where p and q are non-zero constants, has three real roots $\alpha$, $\beta$ and $\gamma$. Given that the interval between $\alpha$ and $\beta$ is p and that the interval between $\beta$ and $\gamma$ is p prove that $8p^{3}+27q=0$. From the discriminant, if two roots are common then $4p^{3}+27q=0$ but I'm stuck on the case with distinct roots.

Answer 1

In the case of distinct roots, $\alpha = \beta \pm p$ and $\gamma = \beta \mp p$:

$$\implies -p = \alpha + \beta + \gamma = (\beta + p) + \beta + (\beta - p) = 3\beta$$ $$\implies \beta = -\frac{p}{3}$$ $$\implies -q = \alpha\beta\gamma = (\beta + p)\beta(\beta - p) = \left(\frac{2p}{3}\right)\left(-\frac{p}{3}\right)\left(-\frac{4p}{3}\right)$$ $$\therefore 8p^3 + 27q = 0$$