1. The problem involves finding the discriminant $\Delta$ and the roots $x_1$ and $x_2$ of a quadratic equation of the form $ax^2 + bx + c = 0$. 2. The formula for the discriminant is: $$\Delta = b^2 - 4ac$$ This helps determine the nature of the roots. 3. The roots are given by the quadratic formula: $$x_1 = \frac{-b + \sqrt{\Delta}}{2a}$$ $$x_2 = \frac{-b - \sqrt{\Delta}}{2a}$$ These are the two solutions where the parabola intersects the x-axis. 4. The graph is a parabola opening upwards, intersecting the x-axis near $(-1, 0)$ and $(4, 0)$, and the y-axis near $(0, -5)$. 5. The vertex is near $(1.5, -7)$, confirming that $a > 0$ (parabola opens upwards) and the minimum value is at the vertex. 6. Based on the given points, approximating the quadratic equation could be: Using vertex form: $$y = a(x - h)^2 + k$$ where vertex $(h, k) = (1.5, -7)$. Using the y-intercept $(0, -5)$: $$-5 = a(0 - 1.5)^2 - 7$$ $$-5 + 7 = a(2.25)$$ $$2 = 2.25a$$ $$a = \frac{2}{2.25} = \frac{8}{9} \approx 0.89$$ 7. The approximate quadratic equation then is: $$y = \frac{8}{9}(x - 1.5)^2 - 7$$ 8. This aligns with the graph and x-intercepts near $-1$ and $4$. Final answer: $$\Delta = b^2 - 4ac$$ $$x_1 = \frac{-b + \sqrt{\Delta}}{2a}$$ $$x_2 = \frac{-b - \sqrt{\Delta}}{2a}$$ and the approximate quadratic function based on the graph is: $$y = \frac{8}{9}(x - 1.5)^2 - 7$$