I'm working on a problem that involves an equation in the form
$$ x^2[A][v] + x[B][v] + [C][v] = [0] $$
where $[A], [B], [C]$ are some known full rank $R^{n\times n}$ square matrices, $[v]$ is some arbitrary $R^n$ vector, and $x$ is some unknown. I know one way to solve this is to find $x$ such that the determinant of the system is zero. However, this does not seem to play nice with the root-finding algorithms I am using, especially for larger matrices with complex values. Therefore, I would instead like to pose this in the form of an eigenvalue problem. I know this can be done easily in the case $[A]$ or $[B]$ is equal to zero, but is there a way to pose this as a generalized eigenvector problem with eigenvalue $x$ when the problem is in the form above?