Quadratic Roots Vertex 246321

1. **State the problem:** Find the roots and the vertex of the quadratic function $$y = -x^2 - 20x - 19$$. 2. **Formula for roots:** The roots of a quadratic $$ax^2 + bx + c = 0$$ are given by the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 3. **Identify coefficients:** Here, $$a = -1$$, $$b = -20$$, and $$c = -19$$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-20)^2 - 4(-1)(-19) = 400 - 76 = 324$$ 5. **Calculate the roots:** $$x = \frac{-(-20) \pm \sqrt{324}}{2(-1)} = \frac{20 \pm 18}{-2}$$ 6. **Calculate each root:** $$x_1 = \frac{20 + 18}{-2} = \frac{38}{-2} = -19$$ $$x_2 = \frac{20 - 18}{-2} = \frac{2}{-2} = -1$$ 7. **Formula for vertex:** The vertex $$ (h, k) $$ of a parabola $$y = ax^2 + bx + c$$ is given by: $$h = -\frac{b}{2a}$$ $$k = f(h) = a h^2 + b h + c$$ 8. **Calculate vertex x-coordinate:** $$h = -\frac{-20}{2(-1)} = \frac{20}{-2} = -10$$ 9. **Calculate vertex y-coordinate:** $$k = -1(-10)^2 - 20(-10) - 19 = -100 + 200 - 19 = 81$$ 10. **Final answers:** Roots: $$-19$$ and $$-1$$ Vertex: $$(-10, 81)$$