How do we go about solving equations of this sort, where we need to find $x$ satisfying the below? Here $K$ and $\xi$ are known constants. Also, $\phi$ and $\mathbf{\Phi}$ are the standard normal PDF and CDF, respectively.
\begin{eqnarray*} \frac{\xi\left(K-x\right)^{2}\phi\left(\xi\left\{ K-x\right\} \right)}{\Phi\left(\xi\left\{ K-x\right\} \right)}+\left(K-x\right)\left[\frac{\phi\left(\xi\left\{ K-x\right\} \right)}{\Phi\left(\xi\left\{ K-x\right\} \right)}\right]^{2} & & =\\ \left\{ K-2x\right\} +\frac{1}{\xi}\frac{\phi\left(\xi\left\{ K-x\right\} \right)}{\Phi\left(\xi\left\{ K-x\right\} \right)}+\frac{\xi Kx\phi\left(\xi x\right)}{\Phi\left(\xi x\right)}+K\left[\frac{\phi\left(\xi x\right)}{\Phi\left(\xi x\right)}\right]^{2} \end{eqnarray*}
This comes up during the minimization of this problem.
\begin{eqnarray*} \underset{\left\{ x\right\} }{\min}\left[K\left\{ \xi x+\frac{\phi\left(\xi x\right)}{\Phi\left(\xi x\right)}\right\} +\left(K-x\right)\left\{ \xi\left(K-x\right)+\frac{\phi\left(\xi\left(K-x\right)\right)}{\Phi\left(\xi\left(K-x\right)\right)}\right\} \right] \end{eqnarray*}
Please note this can be shown to be convex and there is a separate thread on this. https://stats.stackexchange.com/questions/158042/convexity-of-function-of-pdf-and-cdf-of-standard-normal-random-variable