Hello Sydney,
The left side is a "Sum of Cubes".
The identity for factoring that is, generally speaking,
a3 + b3 = (a + b) * (a2 - ab + b2).
So, to emphasize how they compare, I am going to re-write the left side a bit to get
(x)3 + (6)3 = 0. So a = x and b = 6,
→ (x + 6) * (x2 - 6x + 36) = 0
→ x = -6 is a 'real' root.
But we are looking for the complex roots, which will come from the other factor by using either the quadratic formula, or by completing the square.
Using the quadratic formula, you would have
x = [6 ± √(36 - 4(1)(36)] / (2)
→ x = [6 ± √(-108)] / (2)
→ x = [6 ± 6i√(3)] / (2)
→ x = 3 ± 3i√(3)
Thank you for the question, and have a great day.
Michael E.
