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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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up vote 8 down vote accepted

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. :-) –  Chris's sis Jul 4 '12 at 20:05
    
@Chris' sister: It’s a cute problem. –  Brian M. Scott Jul 4 '12 at 20:08
    
do you know other solutions excepting this one? Apparently it's easy, but without this trick one may be in trouble. (i think) –  Chris's sis Jul 4 '12 at 20:15
3  
@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. –  Brian M. Scott Jul 4 '12 at 20:19
    
in fact, yes, i wanted to know how did you think of it, but now it's clear. The approach is very unusual. –  Chris's sis Jul 4 '12 at 20:21

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