Try this beautiful problem from TOMATO Objective no. 258 based on Real Roots of a Cubic Polynomial.
Problem: Real Roots of a Cubic Polynomial
Let a,b,c be distinct real numbers. Then the number of real solution of is
(D) depends on a,b,c
Solution: Ans: (A)
But it is not possible a quadratic equation has three roots.so, it implies that f'(x) has no real roots.But f(x) is a cubic polynomial. And we know a cubic polynomial must have at least one real root ( we know all polynomial curves are continuous ,so it cuts either of the axes at least once).So, number of real root of the given equation is 1 .