1. **State the problem:** Solve the quadratic equation $x^2 - 8x - 20 = 0$. 2. **Formula used:** The quadratic formula is given by $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where the quadratic equation is in the form $ax^2 + bx + c = 0$. 3. **Identify coefficients:** Here, $a = 1$, $b = -8$, and $c = -20$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-8)^2 - 4 \times 1 \times (-20) = 64 + 80 = 144$$ 5. **Apply the quadratic formula:** $$x = \frac{-(-8) \pm \sqrt{144}}{2 \times 1} = \frac{8 \pm 12}{2}$$ 6. **Find the two solutions:** - For the plus sign: $$x = \frac{8 + 12}{2} = \frac{20}{2} = 10$$ - For the minus sign: $$x = \frac{8 - 12}{2} = \frac{\cancel{8 - 12}}{2} = \frac{-4}{2} = -2$$ 7. **Final answer:** The solutions to the equation are $x = 10$ and $x = -2$.