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 =
`(-2 +/- √(4 - 16) )/(2) = (-2 +/- √(-12) )/(2)`
x =
`(-2 +/- 2i √(3) )/(2) = (-1 +/- √(3)i )/(1)`

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