Ukrainian National Mathematical Olympiad - Third Round

Obviously the equation can be solved by trivial transformations and reduction to a quadratic one. We suggest a different approach. Denote the left-hand and right-hand sides by $f(x)$ and $g(x)$ respectively. It's easy to verify that $f(a) = g(a) = 0$ and $f(b) = g(b) = 0$. But this implies that distinct numbers $a$, $b$ are roots of this equation. By comparing the coefficients at $x^3$ for both sides, we conclude that our equation is quadratic. Hence, $x=a$ and $x=b$ are its only roots.