Is there an analytic solution? If not, why so?
?
All this tells me is that
I didn't check your calculations but from the last equation you know that :
is positive
Therefore the denominator of the fraction must be negative
EDIT:
Probably you have noticed that your last equation can't be used to solve it for x because you have the variable on both sides of the equation.
I "solved" (using Newton's method) the equation:
and I got 2 real approximate results: