The roots of the quadratic equation $x^2-35x+300=0$ are $\alpha$ and $\beta$. The quadratic equation which has only one real root $\alpha^3+\beta^3$ can be expressed as $x^2-mx+n=0$, where $m$ and $n$ are positive integers.

Find the sum of digits of $n$ without using a calculator.

**Bonus**: Solve this problem without finding the values of $\alpha$ and $\beta$.