I really need your help to solve this exponential equation. It looks so simple, but I haven't been able to find a solution so far:
$$ {A_1 + A_2 \over 2} = A_1 \exp\left({-x^2 \over c_1^2}\right) + A_2 \exp\left({-x^2 \over c_2^2}\right)$$
$$ x = \cdots$$
As Lubos Motl said it is doubtful in the extreme that you will be able to solve this equation analytically.
Suppose that the numbers $A_i$ are positive, then continuity implies that there is at least one solution. Simply evaluate the right hand side and the left hand side separately at both $x=0$ and infinity. Moreover, your right hand side is rapidly decaying function. This property alone will allow to rapidly find a bracket $[0,b]$ which is certain to contain a root. You can then apply the bisection algorithm and narrow the bracket as much as you like or until rounding errors become a problem.