How could I (numerically) solve this equation for $\alpha$ given $x_i$ (these are known) ?
$\sum_{i=1}^N\frac{1}{x_i-\alpha} = \frac{2N}{\sum_{i=1}^{N}(x_i-\alpha)^2}\sum_{i=1}^{N}{(x_i-\alpha)}$
In fact my aim is to solve the following system for $\alpha$ and $\beta$:
$\beta = \sqrt{\frac{\sum_{i=1}^N(x_i-\alpha)^2}{2N}} (1)$
$\sum_{i=1}^{N}\frac{1}{x_i-\alpha} = \frac{1}{\beta^2}\sum_{i=1}^{N}(x_i-\alpha) (2)$
Thus the very first equation is obtained where I put (1) in (2).
Equations (1) and (2) are obtained while solving for the maximum likelihood estimators the following distribution (special case of Weibull): $f(x) = \frac{x_i-\alpha}{\beta^2}e^{-\frac{(x_i-\alpha)^2}{2\beta^2}}$