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Given the function $$ f:\mathbb{R}\rightarrow \mathbb{R}^{n\times n}\\ f(x)=x^2I +xA+B $$ where $I$ is the identity and $A$ and $B$ are given matrices. For which values of $x$ is the resulting matrix singular?

It would of course be easy if $A$ and $B$ commuted but I would like to have an answer for the most general case possible.

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    Without the $x^2$ term, this is the [generalized eigenvalue problem](https://en.wikipedia.org/wiki/Eigendecomposition_of_a_matrix#Generalized_eigenvalue_problem) – 2017-01-18
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    @Omnomnomnom while that is true I fail to see how that helps in solving the promblem I'm having... – 2017-01-18
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    it doesn't. It does, however, indicate that any solution here is not going to have any "nice form" in the most general case. – 2017-01-18

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