Inspired by Calvin Lin - Part III

Algebra Level 5

(x+17)2+(2x+16)2+(3x+15)2++(16x+2)2+(17x+1)2=0(x+17)^2+(2x+16)^2+(3x+15)^2+\ldots+(16x+2)^2+(17x+1)^2=0

If α\alpha and β\beta are the roots of the equation above, and suppose α+βαβ=ab| \alpha + \beta - \alpha \beta | = \frac a b for coprime positive integers a,ba,b, find the value of a+ba+b.

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