# How do you find all the real and complex roots of f(x)= x^3-27?

Feb 6, 2016

$x = 3 , \frac{- 3 \pm 3 i \sqrt{3}}{2}$

#### Explanation:

The roots occur when the function equals $0$.

${x}^{3} - 27 = 0$

Note that ${x}^{3} - 27$ can be factored as a difference of cubes.

$\left(x - 3\right) \left({x}^{2} + 3 x + 9\right) = 0$

The $\left(x - 3\right)$ term yields a root of $x = 3$.

The roots of $\left({x}^{3} + 3 x + 9\right)$ can be found through the quadratic equation:

$x = \frac{- 3 \pm \sqrt{9 - 36}}{2} = \frac{- 3 \pm \sqrt{- 27}}{2} = \frac{- 3 \pm 3 i \sqrt{3}}{2}$