Find all solutions of the equation in the complex number system. X^3-8=0

Question

basic concept

Answer 1

Begin by factoring the equation.  Note that you have a difference of cubes.

 

(x³ - 8) = (x³ - 2³) = (x - 2)[x² + x(2) + 2² ] = (x - 2)(x² + 2x + 4)

 

 

Note that x = 2 is solution.  Get the other 2 solutions by applying the quadratic equation to x² + 2x + 4.

 

x = (1/2){-2 ± √[2² - 4(4)]} = (1/2)[-2 ± √(-12)] = (1/2)[-2 ± 2√(-3)] = -1 ± i√3

 

The 3 solutions are 2, -1 + i√3, and -1 - i√3.