$$\begin{array}{|c|c|c|c|c|} \hline X&0&200&400&800&1000&2000\\ \hline \mathbb{P}&0.4&0.2&0.1&0.1&0.1&0.1\\ \hline \end{array}$$ The principle of zero utility formula:
\begin{equation} \mathbb{E}u(x+\pi(X)-X)=u(w). \end{equation}
I am given $u(x)=\sqrt{x}$, $w=10000$. Now
\begin{equation} \mathbb{E}\sqrt{10000+\pi(X)-X}=\sqrt{10000}. \end{equation}
I know I cannot just simply square everything and get my result. What is the way to get $\pi(X)$? I found similar question (Quadratic Utility Function) and understand the whole procedure with quadratic utility function.