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Re: Bug in Solve?

• To: mathgroup at smc.vnet.net
• Subject: [mg102951] Re: [mg102921] Bug in Solve?
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Thu, 3 Sep 2009 05:37:34 -0400 (EDT)
• References: <200909020803.EAA03289@smc.vnet.net>

On 2 Sep 2009, at 10:03, tonysin wrote:

> I am just trying to learn Mathematica. What am I doing wrong here?
>
> I have a very simple equation:
>
> x^3 - 15 x + 2 = 0
>
> When I plot it in Mathematica 7,
>
> ClearAll[*]
> f[x_] := x^3 - 15 x + 2
> Plot[f[x], {x, -5, 5}]
>
>
> it gives the expected graph of a cubic, with three real roots near -4,
> 0, and 4.
>
>
> When I NSolve it,
>
> NSolve[f[x] == 0, x]
>
> it gives
>
> {{x -> -3.938}, {x -> 0.133492}, {x -> 3.80451}}
>
> which is exactly what you would expect from the graph.
>
> But when I Solve it
>
> Solve[f[x] == 0, x]
>
> it gives this mess
>
> {{x -> 5/(-1 + 2 I Sqrt[31])^(1/3) + (-1 + 2 I Sqrt[31])^(
>    1/3)}, {x -> -((5 (1 + I Sqrt[3]))/(
>     2 (-1 + 2 I Sqrt[31])^(1/3))) -
>    1/2 (1 - I Sqrt[3]) (-1 + 2 I Sqrt[31])^(1/3)}, {x -> -((
>     5 (1 - I Sqrt[3]))/(2 (-1 + 2 I Sqrt[31])^(1/3))) -
>    1/2 (1 + I Sqrt[3]) (-1 + 2 I Sqrt[31])^(1/3)}}
>
>
> I don't know how it looks in your font, but that "I" in each solution
> is the imaginary i.  Solve is saying this equation has no real roots,
> even though the graph clearly shows that all three roots are real.
>
> Can someone tell me if I am doing something wrong, or am I expecting
> something wrong, or if I just can't trust Mathematica?  Thanks for any
> help.
>

You should learn a little more mathematics. The fact that your  
expressions contain I does not mean at all that they have non-zero  
imaginary parts, only that Mathematica does not attempt by itself to  
find an expression without I automatically (this is a reasonable thing  
to do as trying to find such an expression would take, in general, a  
lot of extra time which would, in most cases, be wasted. But if you  
want a purely real expression then:


FullSimplify[ComplexExpand[Solve[x^3 - 15*x + 2 == 0, x]]]
{{x -> Sqrt[5]*
          (Sqrt[3]*Sin[(1/3)*ArcTan[2*Sqrt[31]]] +
             Cos[(1/3)*ArcTan[2*Sqrt[31]]])},
    {x -> -2*Sqrt[5]*Cos[(1/3)*ArcTan[2*Sqrt[31]]]},
    {x -> Sqrt[5]*(Cos[(1/3)*ArcTan[2*Sqrt[31]]] -
             Sqrt[3]*Sin[(1/3)*ArcTan[2*Sqrt[31]]])}}

gives you one.

Andrzej Kozlowski

• Follow-Ups:
  • Re: Re: Bug in Solve?
    • From: Murray Eisenberg <murray@math.umass.edu>

• References:
  • Bug in Solve?
    • From: tonysin <a2mgoog@yahoo.com>