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I have this equation, and I want to find solution for x.

$\begin{align*}&(-2 x+2 α+1/(2 σ^2))\exp[(-(x-α)^2+(x-μ)/(2 σ^2))]+\\&(-2 x+2 β+1/(2σ^2))\exp[(-(x-β)^2+(x-μ)/(2 σ^2))]=0\end{align*}$

I have already used Wolfram, but it calculates the solution for $\sigma$, and other solvers say that they "Can not solve for x". Does anyone have an idea? Thank you all, in advance, for your concern.

P.S. Can I say that if this equation is equal to $0$, then only $(-2x+2 α+1/(2 σ^2))=0$ and $(-2 x+2 β+1/(2σ^2))=0$ at the same time, since exp is always $>0$.

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    It looks to me that you'd be hard-pressed to find a closed-form solution, given that you have polynomials of different degrees inside and outside the exponential...2011-11-20
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    @J.M. What do you think about my idea at (P.S.)? It's completely wrong?2011-11-20
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    According to Maple solution for $x$ is composed of [Lambert W functions](http://en.wikipedia.org/wiki/Lambert_W_function)2011-11-20
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    If you're sure everything is always real, then the P.S. ought to nail it, yes.2011-11-20
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    No, you cannot say the two polynomials are both $0$. One could be positive and the other negative.2011-11-20
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    @RossMillikan ok you definitely have a point. Do you have any idea how i should proceed?2011-11-20
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    In fact, if we assume $\alpha \lt \beta$, \alpha+1/(4\sigma^2) \lt x \lt \beta+1/(4\sigma^2)$ (Otherwise just flip the inequalities). This will allow numeric solution to proceed easily. You could check out chapter 9 of http://apps.nrbook.com/c/index.html or any numerical analysis text. Like the others, I don't think you will find an algebraic solution.2011-11-20
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    @RossMillikan The fact is, that i must find an algebraic solution for x, in any way, because i want to construct a rejection algorithm then.2011-11-20

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