Find the non real roots of the following equation.

Question

Find the non real roots of the following equation. 

   

x=

Answer 1

Using the identity

a³ - b³ = (a - b)(a² + ab + b²)

we can rewrite

f(x) = x³ - 3³ = (x - 3)(x² + 3x + 9)

Therefore besides the real root 3, f(x) has the roots of

x² + 3x + 9 = 0

whose roots are of the form

x = (-3 +- sqrt(9 - 36))/2

So the two complex root are

x1 = (3/2)(-1 + i sqrt(3))

x2 = (3/2)(-1 - i sqrt(3))