1. **State the problem:** Solve the quadratic equation $$x^2 - 5x - 14 = 0$$. 2. **Formula used:** The quadratic formula is $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where the equation is in the form $$ax^2 + bx + c = 0$$. 3. **Identify coefficients:** Here, $$a = 1$$, $$b = -5$$, and $$c = -14$$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-5)^2 - 4(1)(-14) = 25 + 56 = 81$$. 5. **Find the square root of the discriminant:** $$\sqrt{81} = 9$$. 6. **Apply the quadratic formula:** $$x = \frac{-(-5) \pm 9}{2(1)} = \frac{5 \pm 9}{2}$$. 7. **Calculate the two solutions:** - $$x_1 = \frac{5 + 9}{2} = \frac{14}{2} = 7$$ - $$x_2 = \frac{5 - 9}{2} = \frac{-4}{2} = -2$$. 8. **Final answer:** The solutions to the equation are $$x = 7$$ and $$x = -2$$.