Question

If the ratio of the roots of the quadratic equation $x^2+px+q=0$ be equal to the ratio of the roots of $x^2+lx+m=0$, then prove that $p^{2}m=l^{2}q$.

Answer 1

$x^{2}px+q=0$ $x^{2}lx+m=0$

$\Rightarrow \alpha +\beta =-p$ $\Rightarrow \alpha +\beta =-l$ $\alpha \beta =m$

$\Rightarrow \alpha \beta =q$ $\Rightarrow (\alpha -\beta )=\sqrt{l^2-4m}$

$\Rightarrow \alpha -\beta =\sqrt{p^2-4q}$ $\Rightarrow 2=\frac{-l+\sqrt{l^2-4m}}{2}$