1. **State the problem:** Solve the quadratic equation $$x^2 + 3x - 9 = 0$$. 2. **Formula used:** The quadratic formula to solve $$ax^2 + bx + c = 0$$ is $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$. 3. **Identify coefficients:** Here, $$a = 1$$, $$b = 3$$, and $$c = -9$$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 3^2 - 4 \times 1 \times (-9) = 9 + 36 = 45$$. 5. **Evaluate the roots:** $$x = \frac{-3 \pm \sqrt{45}}{2} = \frac{-3 \pm 3\sqrt{5}}{2}$$. 6. **Final answer:** The solutions are $$x = \frac{-3 + 3\sqrt{5}}{2}$$ and $$x = \frac{-3 - 3\sqrt{5}}{2}$$. These are the two roots of the quadratic equation.