Question

Find all the complex roots. write the answer in rectangular form.the complex cube roots of 8

Answer 1

 x³ = 8 
 x³  -  8 = 0
a=x      b=2

a³ - b³ = (a - b)(a² + ab + b²)
x³  - 8 = (x - 2)(x² + 2x + 2²)

(x - 2)(x² + 2x + 2²) = 0
(x - 2) = 0,    (x² + 2x + 4) = 0
       x = 2      $\frac{-b +/- \sqrt{b x^{2} - 4ac} }{2a} = \frac{-2 +/- \sqrt{2 x^{2} - 4(1)(4)} }{2(1)}$
                     x = $\frac{-2 +/- \sqrt{4 - 16} }{2} = \frac{-2 +/- \sqrt{-12} }{2}$
                     x = $\frac{-2 +/- 2i \sqrt{3} }{2} = \frac{-1 +/- \sqrt{3}i }{1}$

Answer: the complex roots of 8 are: 2, -1 + √3i, -1 - √3i