# 2015 Countdown Problem 20: A Cubic Expansion in a Quadratic Equation

Algebra Level 4

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 $$\beta$$.

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