1. **State the problem:** Solve the quadratic equation $$z^2 - 6z - 63 = 0$$. 2. **Formula used:** For a quadratic equation $$az^2 + bz + c = 0$$, the solutions are given by the quadratic formula: $$z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 3. **Identify coefficients:** Here, $$a = 1$$, $$b = -6$$, and $$c = -63$$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-6)^2 - 4 \times 1 \times (-63) = 36 + 252 = 288$$ 5. **Apply the quadratic formula:** $$z = \frac{-(-6) \pm \sqrt{288}}{2 \times 1} = \frac{6 \pm \sqrt{288}}{2}$$ 6. **Simplify the square root:** $$\sqrt{288} = \sqrt{144 \times 2} = 12\sqrt{2}$$ 7. **Substitute back:** $$z = \frac{6 \pm 12\sqrt{2}}{2}$$ 8. **Simplify the fraction:** $$z = \frac{\cancel{6}^3 \pm \cancel{12}^6\sqrt{2}}{\cancel{2}^1} = 3 \pm 6\sqrt{2}$$ 9. **Final answer:** $$z = 3 + 6\sqrt{2} \quad \text{or} \quad z = 3 - 6\sqrt{2}$$