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# Quadratic Discriminant

This one number can tell you whether the solutions to a quadratic equation are real or non-real, and whether they are distinct or repeated.

Find the number of positive real roots of the equation \[x^{4}-4x-1=0.\]

\(\begin{equation} \displaystyle \prod_{k=1}^{999} (x^2-47x+k) = (x^2-47x+1)(x^2-47x+2)\dots(x^2-47x+999) \end{equation}\)

If the product of all **real** roots of the polynomial above can be expressed in the form \(n!\), what is the value of \(n\)?

Find the number of real roots of the polynomial \(3x^{5} -25x^{3} +60x\).

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