Calculus or Algebra?

If we solve the the equation in $x$ we find two roots:

$x_1=\frac{-I-\sqrt{\Delta}}{2(I-m)}, x_2=\frac{-I+\sqrt{\Delta}}{2(I-m)}, \Delta=I^{2}-4(I-m)$

Without loss of generalities:

$2x_1=x_2$

$2( -I-\sqrt{\Delta} )=-I+\sqrt{\Delta}$

$m=\frac{9I-2I^{2}}{9}$

Now we can study $m(I)$ as a function of $I$ and find a $max$ in $I=\frac{9}{4}$

The solution of the problem is $m(\frac{9}{4})=\frac{9}{8}$

$\boxed{1.125}$