What is the complex solution of x cubed equals - 1?

Answer 1

x3 - 1 = (x - 1)(x2 + x + 1)

so (x3 - 1) will be zero if either (x - 1) is zero or (x2 + x + 1) is zero.

To get the complex solutions, solve the quadratic equation x2 + x + 1 = 0

x2 + x + 1 = (x + 1/2)2 + 3/4 = 0

(x + 1/2)2 = -3/4

So

x + 1/2 = i (square-root-of-3) / 2

or

x + 1/2 = - i (square-root-of-3) / 2

The complex solutions are therefore

(1 / 2) (-1 + i (3)(1/2))

and

(1 / 2) (-1 - i (3)(1/2))

Answer 2

The complex roots are 0.5 +/- i/2*sqrt(3)

or cos(pi/3) +/- i sin(pi/3)where the angles are measured in radians.