1. **State the problem:** Solve the quadratic equation $$x^2 - 3x - 6 = 0$$. 2. **Formula used:** The quadratic formula for solving $$ax^2 + bx + c = 0$$ is $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$. 3. **Identify coefficients:** Here, $$a = 1$$, $$b = -3$$, and $$c = -6$$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-3)^2 - 4(1)(-6) = 9 + 24 = 33$$. 5. **Apply the quadratic formula:** $$x = \frac{-(-3) \pm \sqrt{33}}{2(1)} = \frac{3 \pm \sqrt{33}}{2}$$. 6. **Final answer:** The solutions are $$x = \frac{3 + \sqrt{33}}{2}$$ and $$x = \frac{3 - \sqrt{33}}{2}$$. These are the two roots of the quadratic equation.