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 \(\beta\).

This problem is part of the set 2015 Countdown Problems.

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