Given the equation $Att^T+Bt+C=0$ where $A \in \mathbb{R}^{m\times m\times m}$ and $B \in \mathbb{R}^{m\times m}$ and $C \in \mathbb{R}^m$ and $A,B$ are symmetric, can it be solved for $t \in \mathbb{C}^{m}$ in a closed-form fashion via some variant of the quadratic formula? This comes from an attempt at using a 3rd-order Newton optimization step.
Following the traditional quadratic formula approach, you end up needing the square root of $\frac{1}{4}A^{-1}BA^{-1}B-A^{-1}C$. It's not obvious to me that this value is symmetric; maybe it is. I've read some good ideas for computing the square root of symmetric matrices. Even if the square root was not exact, meaning I had two roots that are approximately equal, that would still be interesting.