1. **State the problem:** Solve the cubic equation $$x^3 - 6x^2 + 6x = 0$$. 2. **Rewrite the equation:** $$x^3 - 6x^2 + 6x = 0$$ 3. **Factor out the common factor $x$:** $$x(x^2 - 6x + 6) = 0$$ 4. **Set each factor equal to zero:** - For $x = 0$ - For the quadratic $x^2 - 6x + 6 = 0$ 5. **Solve the quadratic using the quadratic formula:** The quadratic formula is: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=1$, $b=-6$, and $c=6$. 6. **Calculate the discriminant:** $$b^2 - 4ac = (-6)^2 - 4(1)(6) = 36 - 24 = 12$$ 7. **Find the roots of the quadratic:** $$x = \frac{-(-6) \pm \sqrt{12}}{2(1)} = \frac{6 \pm 2\sqrt{3}}{2} = 3 \pm \sqrt{3}$$ 8. **List all solutions:** $$x = 0, \quad x = 3 + \sqrt{3}, \quad x = 3 - \sqrt{3}$$ **Final answer:** $$\boxed{0, \quad 3 + \sqrt{3}, \quad 3 - \sqrt{3}}$$