1. **State the problem:** Solve the quadratic equation $$x^2 + 27x - 10 = 0$$. 2. **Recall the quadratic formula:** For any quadratic equation $$ax^2 + bx + c = 0$$, the solutions are given by $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$. 3. **Identify coefficients:** Here, $$a = 1$$, $$b = 27$$, and $$c = -10$$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 27^2 - 4 \times 1 \times (-10) = 729 + 40 = 769$$. 5. **Apply the quadratic formula:** $$x = \frac{-27 \pm \sqrt{769}}{2 \times 1} = \frac{-27 \pm \sqrt{769}}{2}$$. 6. **Final answer:** The solutions are $$x = \frac{-27 + \sqrt{769}}{2}$$ and $$x = \frac{-27 - \sqrt{769}}{2}$$. These are the exact roots of the quadratic equation.