Given $$A_{i j k}X_j X_k + B_{ij} X_j + C_i = 0$$ where $A_{ijk}$, $B_{ij}$, and $C_i$ are arbitrary real numbers for all $i$, $j$, $k$ which are $N$-dimensional indices, such that $A_{ijk}=A_{ikj}$ holds. There is a summation for repeated indices, so this is $N$ equations with $N$ unknowns. Is it possible to determine $\vec{X}$ analytically?

If a full solution does not exist, is there a solution for special cases? For example

• if $B_{ij}=0$ for all $i$ and $j$, or

• if $C_{i}=0$ for all $i$, or

• if the $A_{ijk}, B_{ij}, C_i$ constant coefficients are independent of $i$ for each fixed $i$.

If such an expression exists, it would allow one to give a second order Newton-Raphson algorithm.