# Prove or disprove an inequality

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 ... ?
@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