Find the solutions to $z^3 = 2 + 11i$.
I wrote $z = x + iy$ to get $(x + iy)^3 = 2+ 11i$.
Expanding and equating the real and Imaginary parts I got,
$$x^3 - 3xy^2 = 2$$ $$y^3 - 3x^2y = 11$$
Let $tx = y$, Substituting this I got,
$$x^3 - 3t^2x^3 = 2$$ $$x^3(3t-t^3) = 11$$
Susbtituting for $x^3$ from first equation into second eaution I got,
$$2t^3 + -6t - 33t^2 + 11 = 0$$
I don't know how to solve this cubic, I tried finding roots by putting different numbers but I can't find any of them.
What are some other ways of solving without running into a cubic and how can I solve this cubic ?