The original equation is equivalent to a system with $n+1$ unknowns:

$$ \eqalign{ \sum a_i r_i & = 0, \cr r_1^2 - f_1(x) & = 0, \cr &... \cr r_n^2 - f_n(x) & = 0. } $$

I was able to solve such a system with sagemath, so this question should be closed. I understood that myself only after I thought more about the answer by Robert Israel, so it could still be useful for those who don't know about this method. On the other hand, it is strange that an algebraic system could not solve such an equation itself. Please add answers/comments, if there is some more powerful method than this or if I missed something!

The original equation is equivalent to a system with $n+1$ unknowns:

$$ \eqalign{ \sum a_i r_i & = 0, \cr r_1^2 - f_1(x) & = 0, \cr &... \cr r_n^2 - f_n(x) & = 0, } $$

with additional requirements all $r_i \geq 0$.

I was able to solve such system with sagemath (UPD: only for the simplest case, not for the updated question). Please add answers/comments, if there is some more powerful method than this or if I missed something!