What's the median of $f(x) = 4xe^{-2x}$?

Letting $x=q_{50}$, as @Vítězslav Štembera answered, you want to solve for $x$ the equation $$ (2 x+1)\,e^{-2 x}=k \quad\implies\quad(2x+1)\,e^{-(2 x+1)}=\frac k e$$ The only explicit solution of it is given by $$x=-\frac{1}{2} \left(1+W_{-1}\left(-\frac{k}{e}\right)\right)$$ where $W_{-1}(.)$ is the second branch of Lambert function.

If you cannot use Lambert function, only numerical methods would give the solution.

Your equation \begin{align} -2q_{50}e^{-2q_{50}} - e^{-2q_{50}} + 1 = 0.5 \end{align} i.e. \begin{align} (2q_{50}+1)e^{-2q_{50}}= 0.5 \end{align} is correct, however it is trascendental and must be solved numerically. Using MAPLE for example you can find $q_{50}\approx 0.839173495$.