# Why does this say that there are two complex roots, when they are displayed on real axes?

How do I find the real valued solutions to $3x - x^3 = \sqrt{(x + 2)}$.

http://www.wolframalpha.com/input/?i=3x+-+x%5E3+%3D+sqrt%28x+%2B+2%29

Here we see three intersections of the two graphs $f(x) = 3x - x^3$ and $g(x) = \sqrt{(x + 2)}$. I don't see how this is possible since the two complex solutions are "plotted" even though the axes are real-valued.

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## 1 Answer

Look at the approximate forms of the solutions: you'll notice that the imaginary part of the two "complex" solutions is something like $10^{-17}$. I guess that it's numerical noise.

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Then what are the extra two things they are plotting? Why are they on the plots? –  Arthur Collé Oct 27 '12 at 19:23
@A.M.C.: There are indeed three real roots. WolframAlpha is merely expressing them using complex numbers; as Andrea says, the imaginary parts cancel out in the end. Sometimes this is necessary: see casus irreducibilis. –  Rahul Oct 27 '12 at 20:19
If the approximation comes out that low, it's likely that the true value of the imaginary part is $0$. The reason you see $i$ sprinkled throughout the given roots is that the general formula for roots of cubics requires the use of imaginary numbers, even when all roots are real. Presumably a full simplification would reveal an imaginary part of $0$ for all of the roots. –  Tabes Bridges Oct 27 '12 at 20:20