Quadratic Trouble

Algebra Level 4

Let \(f(x) = ax^2 + bx + c\), where \(a,b,c\) are integers. Suppose that \(f(1)=0, 50<f(7)<60, 70<f(8)<80\) and\( 500m<f(100)<500(m+1)\) for some integer \(m\).

Also, let the non-integer root of \(f(x)\) be \(n\). Find \(m+2n\).

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