  The so-called polynomial equations play an important role both in algebra and
in the theory of functional equations. If the unknown functions in the equation
are additive, relatively many results are known. However, even in this case,
there are a lot of open questions. In some specific cases, according to
classical results, the unknown additive functions are homomorphisms,
derivations, or linear combinations of these. The question arises as to whether
the solutions can be described even if the unknown functions are not assumed to
be additive but to be generalized monomials. As a starting point, we will deal
with quadratic functions in this paper. We aim to show that quadratic functions
that are solutions to certain polynomial equations necessarily have a `special'
form. Further, we also present a method to determine these special forms.
