Re: finding roots of 1 + 6*x - 8*x^3

• To: mathgroup at smc.vnet.net
• Subject: [mg54643] Re: [mg54630] finding roots of 1 + 6*x - 8*x^3
• From: Bob Hanlon <hanlonr at cox.net>
• Date: Fri, 25 Feb 2005 01:18:41 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com

```The result that you obtained is real and in squart root form

Solve[1 + 6*x - 8*x^3==0,x]//Simplify

{{x -> (2*2^(1/3) + (2 + 2*I*Sqrt[3])^(2/3))/(4*(1 + I*Sqrt[3])^(1/3))},
{x -> (-(1/2))*((1/2)*(1 + I*Sqrt[3]))^(2/3) + (1/4)*I*((1/2)*(1 +
I*Sqrt[3]))^(1/3)*
(I + Sqrt[3])}, {x -> (-2 + 2*I*Sqrt[3] - 2^(1/3)*(1 + I*Sqrt[3])^(5/3))/
(4*2^(2/3)*(1 + I*Sqrt[3])^(1/3))}}

%//N//Chop

{{x -> 0.9396926207859084}, {x -> -0.766044443118978}, {x ->
-0.17364817766693055}}

Reduce[1 + 6*x - 8*x^3==0,x]//N

x == -0.766044443118978 || x == -0.17364817766693036 || x ==
0.9396926207859084

Bob Hanlon

>
> From: "Kennedy" <kennedy at oldnews.org>
To: mathgroup at smc.vnet.net
> Date: 2005/02/24 Thu AM 03:21:35 EST
> To: mathgroup at smc.vnet.net
> Subject: [mg54643] [mg54630] finding roots of 1 + 6*x - 8*x^3
>
> Hello All,
>
> I am trying to find the roots of
> 1 + 6*x - 8*x^3.
>
> Roots[1+6*x-8*x^3==0,x] yields this ugly thing:
> (made uglier by my converting to InputForm)
>
> x == ((1 + I*Sqrt[3])/2)^(1/3)/2 +
>    1/(2^(2/3)*(1 + I*Sqrt[3])^(1/3)) ||
>  x == -((1 - I*Sqrt[3])*((1 + I*Sqrt[3])/2)^(1/3))/4 -
>    ((1 + I*Sqrt[3])/2)^(2/3)/2 ||
>  x == -(1 - I*Sqrt[3])/(2*2^(2/3)*(1 + I*Sqrt[3])^
>       (1/3)) - (1 + I*Sqrt[3])^(4/3)/(4*2^(1/3))
>
> This monstrosity is chock full of imaginaries,
> even though I know all three roots are real.
>
> I tried Solve too but got the same thing, except
> in the form of a set of replacements. My guess is
> that Solve just calls Roots when handed a poly-
> nomial.
>
> When I ran the above through FullSimplify, I
> got three "Root" objects, the upshot of which is
> that the roots of the polynomial are indeed the
> Roots of said polynomial. Huh.
>
> What command can I use to get the roots into
> a form that are
> (a)  purely real, and
>
> Thanks,
> Kennedy
>
> PS.  Mathematica 4.2
>
>

```

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