The equation given is $x^3 - 8 = 0$.

To solve for $x$, follow these steps:

1. Factor the equation: $x^3 - 8 = (x - 2)(x^2 + 2x + 4) = 0$

2. Solve for $x$:

   • For the factor $x - 2 = 0$: $x = 2$
   • For the quadratic $x^2 + 2x + 4 = 0$, use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ Here, $a = 1$, $b = 2$, and $c = 4$.

3. Calculate the discriminant: $b^2 - 4ac = 2^2 - 4 \times 1 \times 4 = 4 - 16 = -12$

4. Find the imaginary solutions: $x = \frac{-2 \pm \sqrt{-12}}{2}$ $x = \frac{-2 \pm 2i\sqrt{3}}{2}$ $x = -1 \pm i\sqrt{3}$

Thus, the solutions are:

• Real solution: $x = 2$
• Imaginary solutions: $x = -1 + i\sqrt{3}$ and $x = -1 - i\sqrt{3}$