       Re: cubic equation solver

• To: mathgroup at smc.vnet.net
• Subject: [mg130248] Re: cubic equation solver
• From: Bob Hanlon <hanlonr357 at gmail.com>
• Date: Thu, 28 Mar 2013 11:55:58 -0400 (EDT)
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• References: <20130328080639.90E5769F4@smc.vnet.net>

```eqn = x^3 + (Sqrt + 2 Sqrt + 2 Sqrt - 9) x +
2 Sqrt - Sqrt - 2 == 0;

sol = Solve[eqn, x] // Simplify

{{x -> (-2*3^(1/3)*(-9 + 2*Sqrt + 2*Sqrt +
Sqrt) + 2^(1/3)*
(9*(2 + Sqrt - 2*Sqrt) +
I*Sqrt[6*(2331 - 626*Sqrt -
648*Sqrt - 132*Sqrt)])^(2/3))/
(6^(2/3)*(9*(2 + Sqrt - 2*Sqrt) +
I*Sqrt[6*(2331 - 626*Sqrt - 648*Sqrt -
132*Sqrt)])^(1/3))},
{x -> (2*3^(1/3)*((-9 - 6*I) + (2 - 3*I)*Sqrt +
(2 + 9*I)*Sqrt + (1 - 2*I)*Sqrt) +
2^(1/3)*(-1 - I*Sqrt)*
(9*(2 + Sqrt - 2*Sqrt) +
I*Sqrt[6*(2331 - 626*Sqrt -
648*Sqrt - 132*Sqrt)])^(2/3))/
(2*6^(2/3)*(9*(2 + Sqrt - 2*Sqrt) +
I*Sqrt[6*(2331 - 626*Sqrt - 648*Sqrt -
132*Sqrt)])^(1/3))},
{x -> (2*3^(1/3)*((-9 + 6*I) + (2 + 3*I)*Sqrt +
(2 - 9*I)*Sqrt + (1 + 2*I)*Sqrt) +
I*2^(1/3)*(I + Sqrt)*
(9*(2 + Sqrt - 2*Sqrt) +
I*Sqrt[6*(2331 - 626*Sqrt -
648*Sqrt - 132*Sqrt)])^(2/3))/
(2*6^(2/3)*(9*(2 + Sqrt - 2*Sqrt) +
I*Sqrt[6*(2331 - 626*Sqrt - 648*Sqrt -
132*Sqrt)])^(1/3))}}

Although not in the simplest form, these are the correct roots

eqn /. sol // Simplify

{True, True, True}

To find simpler forms without using FullSimplify (which seems to go on
indefinitely)

sol2 = ({x -> RootApproximant[#[[-1, -1]]]} & /@ sol) // ToRadicals

{{x -> Sqrt[5 - 2*Sqrt]}, {x -> 2 - Sqrt},
{x -> -2 + Sqrt}}

Verifying that these are exact solutions

eqn /. sol2 // FullSimplify

{True, True, True}

I assume that these are the values in your first sentence which was
garbled (presumably you did not convert to InputForm before copying).

Bob Hanlon

On Thu, Mar 28, 2013 at 4:06 AM, Elim Qiu <elim.qiu at gmail.com> wrote:
> x^3 + (=E2=88=9A6 + 2=E2=88=9A3 + 2=E2=88=9A2 -9)x + 2=E2=88=9A3 -=E2=88=9A2 -2 = 0
> has exact roots =E2=88=9A2-2, =E2=88=9A3-=E2=88=9A2, 2-=E2=88=9A3
>
> But Mathematica says:
>
> Solve[x^3 + (Sqrt + 2 Sqrt + 2 Sqrt - 9) x + 2 Sqrt -
>    Sqrt - 2 == 0, x]
>
> {{x -> (1/
>       2 (18 + 9 Sqrt - 18 Sqrt +
>         I Sqrt[3 (4662 - 1252 Sqrt - 1296 Sqrt -
>             264 Sqrt)]))^(1/3)/3^(
>     2/3) - (-9 + 2 Sqrt + 2 Sqrt + Sqrt[
>      6])/(3/2 (18 + 9 Sqrt - 18 Sqrt +
>         I Sqrt[3 (4662 - 1252 Sqrt - 1296 Sqrt -
>             264 Sqrt)]))^(
>     1/3)}, {x -> -(((1 + I Sqrt) (1/
>         2 (18 + 9 Sqrt - 18 Sqrt +
>           I Sqrt[3 (4662 - 1252 Sqrt - 1296 Sqrt -
>               264 Sqrt)]))^(1/3))/(
>      2 3^(2/3))) + ((1 - I Sqrt) (-9 + 2 Sqrt + 2 Sqrt +
>        Sqrt))/(
>     2^(2/3) (3 (18 + 9 Sqrt - 18 Sqrt +
>          I Sqrt[3 (4662 - 1252 Sqrt - 1296 Sqrt -
>              264 Sqrt)]))^(
>      1/3))}, {x -> -(((1 - I Sqrt) (1/
>         2 (18 + 9 Sqrt - 18 Sqrt +
>           I Sqrt[3 (4662 - 1252 Sqrt - 1296 Sqrt -
>               264 Sqrt)]))^(1/3))/(
>      2 3^(2/3))) + ((1 + I Sqrt) (-9 + 2 Sqrt + 2 Sqrt +
>        Sqrt))/(
>     2^(2/3) (3 (18 + 9 Sqrt - 18 Sqrt +
>          I Sqrt[3 (4662 - 1252 Sqrt - 1296 Sqrt -
>              264 Sqrt)]))^(1/3))}}
>

```

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