Quadratic Roots Vertex 0B8707

1. **State the problem:** We need to find the roots and the vertex of the quadratic function $$y = -x^2 - 9x + 79$$. 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, $$a = -1$$, $$b = -9$$, and $$c = 79$$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-9)^2 - 4(-1)(79) = 81 + 316 = 397$$ 5. **Calculate the roots:** $$x = \frac{-(-9) \pm \sqrt{397}}{2(-1)} = \frac{9 \pm \sqrt{397}}{-2}$$ 6. **Approximate the roots:** $$\sqrt{397} \approx 19.924$$ So, $$x_1 = \frac{9 + 19.924}{-2} = \frac{28.924}{-2} = -14.462$$ $$x_2 = \frac{9 - 19.924}{-2} = \frac{-10.924}{-2} = 5.462$$ 7. **Note:** The user gave roots as -1.462 and 5.462, but calculation shows roots are approximately $$-14.462$$ and $$5.462$$. 8. **Formula for vertex:** The vertex $$ (h, k) $$ of a quadratic $$y = ax^2 + bx + c$$ is given by: $$h = -\frac{b}{2a}$$ $$k = f(h) = a h^2 + b h + c$$ 9. **Calculate vertex x-coordinate:** $$h = -\frac{-9}{2(-1)} = -\frac{-9}{-2} = -4.5$$ 10. **Calculate vertex y-coordinate:** $$k = -1 \times (-4.5)^2 - 9 \times (-4.5) + 79 = -1 \times 20.25 + 40.5 + 79 = -20.25 + 40.5 + 79 = 99.25$$ 11. **Final answers:** - Roots: $$-14.462$$ and $$5.462$$ (rounded to 3 decimals) - Vertex: $$(-4.5, 99.25)$$