Quadratic Binomial Fusion

Algebra Level 5

1+r=110[3r(10r)+r(10r)]=210(α45+β)\displaystyle 1 + \sum_{r = 1}^{10} \left [ 3^r \binom{10}{r} + r \binom{10}{r} \right ] = 2^{10} \left (\alpha \cdot 4^5 + \beta \right )

Consider the above summation, where α,βN\alpha, \beta \in \mathbb N and f(x)=x22xk2+1f(x) = x^2 - 2x -k^2 + 1.

If α,β\alpha, \beta lies strictly in between the roots of f(x)=0f(x) = 0, then find the smallest positive integral value of kk.

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