We consider quadratic functions f that satisfy the additional equation y2 f(x) = x2 f(y) for the pairs $${ (x,y) \in \mathbb{R}^2}$$ that fulfill the condition P(x, y) = 0 for some fixed polynomial P of two variables. If P(x, y) = ax + by + c with $${ a , b , c \in \mathbb{R}}$$ and $${(a^2 + b^2)c \neq 0}$$ or P(x,y) = x n − y with a natural number $${n \geq 2}$$ , we prove that f(x) = f(1) x2 for all $${x \in \mathbb{R}}$$ . Some related problems, admitting quadratic functions generated by derivations, are considered as well.