OFFSET
0,1
COMMENTS
The other two roots are (w1*(27 + 3*sqrt(105))^(1/3) + (27 - 3*sqrt(105))^(1/3))/3 = -0.3854584985... + 1.5638845105...*i, and its complex conjugate, where w1 = (-1 + sqrt(3)*i)/2 = exp(2*Pi*i/3) is one of the complex roots of x^3 - 1.
Using hyperbolic functions these roots are -(1/3)*sqrt(6)*(sinh((1/3)* arcsinh((3/4)*sqrt(6))) + sqrt(3)*cosh((1/3)*arcsinh((3/4)*sqrt(6)))*i), and its complex conjugate.
FORMULA
r = (1/3)*(27 + 3*sqrt(105))^(1/3) - 2/(27 + 3*sqrt(105))^(1/3).
r = ((27 + 3*sqrt(105))^(1/3)+ w1*(27 - 3*sqrt(105))^(1/3))/3, where w1 = (-1 + sqrt(3)*i)/2 = exp(2*Pi*i/3) is one of the complex roots of x^3 - 1.
r = (2/3)*sqrt(6)*sinh((1/3)*arcsinh((3/4)*sqrt(6))).
EXAMPLE
0.770916997059248100825146369307026967255053119363328615100598492976735103...
MATHEMATICA
RealDigits[x /. FindRoot[x^3 + 2*x - 2, {x, 1}, WorkingPrecision -> 120]][[1]] (* Amiram Eldar, Sep 21 2022 *)
PROG
(PARI) solve(x=0, 1, x^3 + 2*x - 2) \\ Michel Marcus, Sep 23 2022
(PARI) polrootsreal(x^3 + 2*x - 2)[1] \\ Charles R Greathouse IV, Sep 30 2022
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, Sep 20 2022
STATUS
approved