What method would I use for this proof?

Question

Show there are no integer, $x$ and $y$, that satisfy $x^{2} + 3y^{2} = 8$. I have no idea where to start unfortunately or what kind of method to start off with.

Answer 1

What is the graph of this formula? Looking at the shape, there aren't a whole lot of integer candidates.

Answer 2

Some people call this a "proof by exhaustion". As noted earlier, $x$ must be between $-2$ and $2$ and $y$ has an even smaller range, so just try all the possible values for $x$ and $y$. Some of my students call this a "proof by Duh!".