2015 Countdown Problem 20: A Cubic Expansion in a Quadratic Equation
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 \(\beta\).