1. **State the problem:** Find the roots and vertex of the quadratic function $$y = -x^2 - 16x + 36$$. 2. **Formula for roots:** Use the quadratic formula $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $$a = -1$$, $$b = -16$$, and $$c = 36$$. 3. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-16)^2 - 4(-1)(36) = 256 + 144 = 400$$. 4. **Calculate the roots:** $$x = \frac{-(-16) \pm \sqrt{400}}{2(-1)} = \frac{16 \pm 20}{-2}$$ 5. **Evaluate each root:** - For the plus sign: $$x = \frac{16 + 20}{-2} = \frac{36}{-2} = -18$$ - For the minus sign: $$x = \frac{16 - 20}{-2} = \frac{-4}{-2} = 2$$ 6. **Formula for vertex:** The vertex $$x$$-coordinate is $$x = -\frac{b}{2a}$$. 7. **Calculate vertex $$x$$-coordinate:** $$x = -\frac{-16}{2(-1)} = \frac{16}{-2} = -8$$ 8. **Calculate vertex $$y$$-coordinate:** Substitute $$x = -8$$ into the original equation: $$y = -(-8)^2 - 16(-8) + 36 = -64 + 128 + 36 = 100$$ 9. **Final answers:** - Roots: $$-18$$ and $$2$$ - Vertex: $$(-8, 100)$$