```Date: Mar 28, 2013 5:12 PM
Author: Bob Hanlon
Subject: Re: cubic equation solver

eqn = x^3 + (Sqrt[6] + 2 Sqrt[3] + 2 Sqrt[2] - 9) x +    2 Sqrt[3] - Sqrt[2] - 2 == 0;sol = Solve[eqn, x] // Simplify{{x -> (-2*3^(1/3)*(-9 + 2*Sqrt[2] + 2*Sqrt[3] +                 Sqrt[6]) + 2^(1/3)*              (9*(2 + Sqrt[2] - 2*Sqrt[3]) +                   I*Sqrt[6*(2331 - 626*Sqrt[2] -                            648*Sqrt[3] - 132*Sqrt[6])])^(2/3))/         (6^(2/3)*(9*(2 + Sqrt[2] - 2*Sqrt[3]) +                 I*Sqrt[6*(2331 - 626*Sqrt[2] - 648*Sqrt[3] -                          132*Sqrt[6])])^(1/3))},   {x -> (2*3^(1/3)*((-9 - 6*I) + (2 - 3*I)*Sqrt[2] +                 (2 + 9*I)*Sqrt[3] + (1 - 2*I)*Sqrt[6]) +            2^(1/3)*(-1 - I*Sqrt[3])*              (9*(2 + Sqrt[2] - 2*Sqrt[3]) +                   I*Sqrt[6*(2331 - 626*Sqrt[2] -                            648*Sqrt[3] - 132*Sqrt[6])])^(2/3))/         (2*6^(2/3)*(9*(2 + Sqrt[2] - 2*Sqrt[3]) +                 I*Sqrt[6*(2331 - 626*Sqrt[2] - 648*Sqrt[3] -                          132*Sqrt[6])])^(1/3))},   {x -> (2*3^(1/3)*((-9 + 6*I) + (2 + 3*I)*Sqrt[2] +                 (2 - 9*I)*Sqrt[3] + (1 + 2*I)*Sqrt[6]) +            I*2^(1/3)*(I + Sqrt[3])*              (9*(2 + Sqrt[2] - 2*Sqrt[3]) +                   I*Sqrt[6*(2331 - 626*Sqrt[2] -                            648*Sqrt[3] - 132*Sqrt[6])])^(2/3))/         (2*6^(2/3)*(9*(2 + Sqrt[2] - 2*Sqrt[3]) +                 I*Sqrt[6*(2331 - 626*Sqrt[2] - 648*Sqrt[3] -                          132*Sqrt[6])])^(1/3))}}Although not in the simplest form, these are the correct rootseqn /. sol // Simplify{True, True, True}To find simpler forms without using FullSimplify (which seems to go onindefinitely)sol2 = ({x -> RootApproximant[#[[-1, -1]]]} & /@ sol) // ToRadicals{{x -> Sqrt[5 - 2*Sqrt[6]]}, {x -> 2 - Sqrt[3]},   {x -> -2 + Sqrt[2]}}Verifying that these are exact solutionseqn /. sol2 // FullSimplify{True, True, True}I assume that these are the values in your first sentence which wasgarbled (presumably you did not convert to InputForm before copying).Bob HanlonOn Thu, Mar 28, 2013 at 4:06 AM, Elim Qiu <elim.qiu@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[6] + 2 Sqrt[3] + 2 Sqrt[2] - 9) x + 2 Sqrt[3] ->    Sqrt[2] - 2 == 0, x]>> {{x -> (1/>       2 (18 + 9 Sqrt[2] - 18 Sqrt[3] +>         I Sqrt[3 (4662 - 1252 Sqrt[2] - 1296 Sqrt[3] ->             264 Sqrt[6])]))^(1/3)/3^(>     2/3) - (-9 + 2 Sqrt[2] + 2 Sqrt[3] + Sqrt[>      6])/(3/2 (18 + 9 Sqrt[2] - 18 Sqrt[3] +>         I Sqrt[3 (4662 - 1252 Sqrt[2] - 1296 Sqrt[3] ->             264 Sqrt[6])]))^(>     1/3)}, {x -> -(((1 + I Sqrt[3]) (1/>         2 (18 + 9 Sqrt[2] - 18 Sqrt[3] +>           I Sqrt[3 (4662 - 1252 Sqrt[2] - 1296 Sqrt[3] ->               264 Sqrt[6])]))^(1/3))/(>      2 3^(2/3))) + ((1 - I Sqrt[3]) (-9 + 2 Sqrt[2] + 2 Sqrt[3] +>        Sqrt[6]))/(>     2^(2/3) (3 (18 + 9 Sqrt[2] - 18 Sqrt[3] +>          I Sqrt[3 (4662 - 1252 Sqrt[2] - 1296 Sqrt[3] ->              264 Sqrt[6])]))^(>      1/3))}, {x -> -(((1 - I Sqrt[3]) (1/>         2 (18 + 9 Sqrt[2] - 18 Sqrt[3] +>           I Sqrt[3 (4662 - 1252 Sqrt[2] - 1296 Sqrt[3] ->               264 Sqrt[6])]))^(1/3))/(>      2 3^(2/3))) + ((1 + I Sqrt[3]) (-9 + 2 Sqrt[2] + 2 Sqrt[3] +>        Sqrt[6]))/(>     2^(2/3) (3 (18 + 9 Sqrt[2] - 18 Sqrt[3] +>          I Sqrt[3 (4662 - 1252 Sqrt[2] - 1296 Sqrt[3] ->              264 Sqrt[6])]))^(1/3))}}>
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