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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.

Level 3

         

\(f(x)=ax^2+bx+c\). \(a,b,c\) are real numbers. Given that \(c(a+b+c)<0\) , then what can we say about \(b^2-4ac\) ?

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

Given a function \[f(x) = ax^{2}-bx-16\] doesn't has 2 distinct real roots. Find the maximum value of \(4a-b\).

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

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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