If the roots of the equation ${\mathrm{px}}^2+2\mathrm{qx}+\mathrm{r}=0$ and ${\mathrm{qx}}^2-2\sqrt{\mathrm{pr}}\mathrm{x}+\mathrm{q}=0$ be real, then

Equations ${\mathrm{px}}^2+2\mathrm{qx}+\mathrm{r}=0$ and ${\mathrm{qx}}^2-2\left(\sqrt{\mathrm{pr}}\right)\mathrm{x}+\mathrm{q}=0$ have real roots, then from first

$$\begin{gathered} 4{\mathrm{q}}^2-4\mathrm{pr}\ge 0\Rightarrow {\mathrm{q}}^2-\mathrm{pr}\ge 0\Rightarrow {\mathrm{q}}^2\ge \mathrm{pr} ...\left[\mathrm{i}\right] \\ \mathrm{and} \mathrm{from} \mathrm{second} 4\left(\mathrm{pr}\right)-4{\mathrm{q}}^2\ge 0 (\mathrm{for} \mathrm{real} \mathrm{root}) \\ \Rightarrow \mathrm{pr}\ge {\mathrm{q}}^2 ...\left[\mathrm{ii}\right] \\ \mathrm{From} \left[\mathrm{i}\right] \mathrm{and} \left[\mathrm{ii}\right], \mathrm{we} \mathrm{get} \mathrm{result} {\mathrm{q}}^2=\mathrm{pr}. \end{gathered}$$