#### Answer

$x_{1} = \sqrt{31}$ and $x_{2} = -\sqrt{31}$

#### Work Step by Step

Given \begin{gather*} x^2-31=0 \end{gather*}
Applying difference of two squares, we have:
\begin{gather*}
(x+\sqrt{31})(x-\sqrt{31})=0
\end{gather*}
Therefore the solutions are $x_{1} = \sqrt{31}$ and $x_{2} = -\sqrt{31}$