The equation x³ – 3qx + 2r = 0 has a repeated root then

${\mathrm{f}}^1\left(\mathrm{x}\right)=3{\mathrm{x}}^2-3\mathrm{q}=0\Rightarrow q^{1/2}$ is a repeated root ;
$$\begin{gathered} q^{1/2}{q}^{1/2}+\alpha q^{1/2}+q^{1/2}\alpha =-3q \\ 2\alpha q^{1/2}=-4q\Rightarrow \alpha =-2q^{1/2} \end{gathered}$$

$q^{1/2},q^{1/2},-2{\mathrm{q}}^{1/2}$ are roots
$\Rightarrow \text{ product of roots }-2{\mathrm{q}}^{3/2}=-2\mathrm{r}\Rightarrow {\mathrm{q}}^3={\mathrm{r}}^2$