Problem

Source: VII International Festival of Young Mathematicians Sozopol, Theme for 10-12 grade

Tags: algebra, function



For a quadratic trinomial $f(x)$ and the different numbers $a$ and $b$ it is known that $f(a)=b$ and $f(b)=a$. We call such $a$ and $b$ conjugate for $f(x)$. Prove that $f(x)$ has no other conjugate numbers.