If f(x)=[x]^{2}-30[x]+221=0 where [x] is the greatest integer function, then what is the sum of <strong>ALL</strong> integer solutions?

Explanation

Let [x] = y. The equation y^2 - 30y + 221 = 0 factors into (y-13)(y-17) = 0. Thus, [x] = 13 or [x] = 17. The integer values of x that satisfy these are exactly x = 13 and x = 17. Their sum is 13 + 17 = 30.

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