Given equations are \( \Large px^{2}+2qx+r=0 \) and \( \Large qx^{2}-2\sqrt{pr}x+q=0 \)

Since, they have real roots

\( \Large 4q^{2}-4pr\ge 0 => q^{2}\ge pr..\) ...(i)

and from second, \( \Large 4 \left(pr\right)-4q^{2}\ge 0 \)

=> \( \Large pr\ge q^{2} \)

From equations (i) and (ii) we get

\( \Large q^{2}=pr \)