Consider this example. I am trying to match the mean and variance of a process using a trinomial tree. This gives me two equations for values $\{x_1, x_2, x_3\} $.
$\begin{equation} \alpha_1x_1 + \alpha_2 x_2 + \alpha_3 x_3 = \mu \end{equation}$
$\begin{equation} \alpha_1x_1^2 + \alpha_2 x_2^2 + \alpha_3 x_3^2 = \sigma^2 + \mu^2 \end{equation}$
such that $\sigma>0$.
Even in the cases where $\mu$ is in the interior of the convex hull of $\{x_1, x_2, x_3\}$, and $\sigma^2 + \mu^2$ is in the interior of the convex hull of $\{x_1^2, x_2^2, x_3^2\}$, I am receiving some solutions where not all values are positive.
Is there a way to check for the condition that the solution is positive.