Complex Cubic Equation z^3+3z+2i=0

Question

How we can solve the equation $z^3+3z+2i=0$ ? And is there exist a general method to solve similar equation?

Answer 1

One way to do this, since $z$ appears only to odd powers, is to set $z=iw$ which enables $i$ to be cancelled and gives you integer coefficients.

Another way is to spot a solution.

The general methods for solving a cubic also work with complex coefficients.

Answer 2

Here is my solution: We have $$z^3+3z+2i=z(z^2+1)+2(z+i)=z(z+i)(z-i)+2(z+i)=(z+i)(z^2-iz+2)=(z+i)^2(z-2i)=0$$ so our equation has three roots $-i,-i,2i$.