The quadratic equation $\displaystyle x^2+Lx+M$ has one root which is twice the other.
a)Prove that $\displaystyle 2L^2=9M$
b)Prove also that the roots are rational whenever L is rational
First the roots are:
$\displaystyle r_1=\frac{-L+\sqrt{L^2-4M}}{2}$ and $\displaystyle r_2=\frac{-L-\sqrt{L^2-4M}}{2}$
So either $\displaystyle r_1=2r_2$ or $\displaystyle r_2=2r_1$ one of these has complex roots so is impossible for this problem, and the other should lead to the required solution.
CB