Can't Factor This Fractional Part

Algebra Level 5

Let \(x\) be the smallest positive real satisfying \[\{x^2\}-2\{x\}+1=0\]

Find the value of \[\lfloor 1000\{x\}\rfloor\]

Details and Assumptions

\(\{n\}\) is the fractional part of \(n\). That is, \(\{n\}=n-\lfloor n\rfloor\) for all positive real \(n\).

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