I have following equation to solve:
$ (\vec{b} + \vec{c}x )^T \mathbb{W} ( \vec{a} + (\vec{b}+\vec{c}x) x ) = 0 $
where $\vec{a},\vec{b}$ and $\vec{c}$ are vectors and $\mathbb{W}$ is a symmetric matrix, and $x$ is just a scalar. I am now seeking the solution for $x$.
This problem appears, when performing analytically a polynomial fit. In the simplified case, where $c$ is not present, the solution is for instance: $x = \frac{-b^TWa}{b^TWb}$
For above's cubic equation, there is of course the general solution for a cubic polynomial, when expanding it to something like $0=\alpha+\beta x+\gamma x^2 + \delta x^3$, but I am hoping for a more compact solution.
Note that while matrix multiplication is not commutative, the multiplication with the scalar $x$ is. This means you can multiply out all your brackets and then multiply out the vectors and matrices. Then this equation becomes a 3rd order polynomial with scalar coefficients. These can be solved algebraically (but it is tedious).