jmc

First, we start by squaring both sides of the equation $$\begin{array}{rl}(\sqrt{2x^2+1}{)}^2& =(\sqrt{19}{)}^2\\ 2x^2+1& =19\\ \Rightarrow 2x^2& =18\\ \Rightarrow x^2& =9\end{array}$$From here, we can see that the only possible values of $x$ are 3 and -3. Therefore the average is $\boxed{0}$.