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Let's consider the following equation where $m,n$ are real numbers:

$$ x^3+mx+n=0 $$

I need to prove/disprove without calculus that for any real root of the above equation we have that: $$ m^2-4 x_1 n \ge 0$$

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Suppose that $x_1$ is a real root of the cubic, and consider the quadratic equation $x_1x^2+mx+n=0$. This must have $x_1$ as a real solution, so ... ?

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    that's a nice trick. :-)2012-07-04
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    @Chris' sister: It’s a cute problem.2012-07-04
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    do you know other solutions excepting this one? Apparently it's easy, but without this trick one may be in trouble. (i think)2012-07-04
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    @Chris' sister: I can’t at the moment think of another. If you’re wondering how I came up with it, I looked at the expression $m^2-4x_1n$ and almost immediately thought *discriminant of quadratic*. Then everything just fell into place.2012-07-04
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    in fact, yes, i wanted to know how did you think of it, but now it's clear. The approach is very unusual.2012-07-04