I'm interested in the vector equation $$ a-Bx-2Cx-\left[\begin{array}{c} x'C^{-1}A_{1}C^{-1}x\\ \vdots\\ x'C^{-1}A_{n}C^{-1}x \end{array}\right]=0 $$ where $x$ is an unknown $n\times 1$ column vector, and the square and symmetric matrices $A_i$, $B$ and $C$ are positive definite. $a$ is also an $n\times 1$ column vector.
Is there any hope of solving for $x$?
What I have tried so far
Let $d_i=x'C^{-1}A_{i}C^{-1}x$, then we can write that system as $$ a-Bx-2Cx-d=0 $$ and its solution is $$ x=\left(B+2C\right)^{-1}\left(a-d\right). $$ From there we can go back to $d$ and find $$ d_{i}=\left(a-d\right)'\left(B+2C\right)^{-1}C^{-1}A_{i}C^{-1}\left(B+2C\right)^{-1}\left(a-d\right). $$ But then we have another quadratic system in $d$ to solve, which does not seem much easier.
Are there well-know solutions for systems of the form $$ d_{i}=\left(a-d\right)'E_{i}\left(a-d\right) $$ for all $i$?