If I have two linear equations, $ax + by = 0$ and $cx + dy = 0$, and I wanted to find out if they had any non-trivial solutions, I would simply check if $(a,b)$ and $(c,d)$ are linearly dependent.

Now suppose I set two quadratic forms equal to zero, that is, $xAx^T=0$ and $xBx^T=0$. This is no longer a linear situation, but are there any similar criteria for checking whether or not there exists a non-trivial solution satisfying both forms, if I let $x$ be a vector over $\mathbb{C}$?

I'm not too fussed about what the solutions are, as long as I can be sure that at least one exists.