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

Algebra Level 4

The roots of the quadratic equation x235x+300=0x^2-35x+300=0 are α\alpha and β\beta. The quadratic equation which has only one real root α3+β3\alpha^3+\beta^3 can be expressed as x2mx+n=0x^2-mx+n=0, where mm and nn are positive integers.

Find the sum of digits of nn without using a calculator.

Bonus: Solve this problem without finding the values of α\alpha and β\beta.

This problem is part of the set 2015 Countdown Problems.

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