1. **Problem statement:** Solve the system of equations graphically: $$y = 2x^2 - 3x - 7$$ and $$y = 2x - 1$$ Also, find the minimum value of the quadratic function $$2x^2 - 3x - 7$$. 2. **Graphical method to solve the system:** The solutions are the points where the parabola and the line intersect. Set the two expressions for $$y$$ equal: $$2x^2 - 3x - 7 = 2x - 1$$ 3. **Rearrange the equation:** $$2x^2 - 3x - 7 - 2x + 1 = 0$$ $$2x^2 - 5x - 6 = 0$$ 4. **Solve the quadratic equation:** Use the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $$a=2$$, $$b=-5$$, $$c=-6$$. 5. **Calculate the discriminant:** $$\Delta = (-5)^2 - 4 \times 2 \times (-6) = 25 + 48 = 73$$ 6. **Find the roots:** $$x = \frac{5 \pm \sqrt{73}}{4}$$ 7. **Approximate the roots:** $$\sqrt{73} \approx 8.544$$ $$x_1 = \frac{5 + 8.544}{4} = \frac{13.544}{4} = 3.386$$ $$x_2 = \frac{5 - 8.544}{4} = \frac{-3.544}{4} = -0.886$$ 8. **Find corresponding y-values:** Use $$y = 2x - 1$$: For $$x_1 = 3.386$$: $$y_1 = 2(3.386) - 1 = 6.772 - 1 = 5.772$$ For $$x_2 = -0.886$$: $$y_2 = 2(-0.886) - 1 = -1.772 - 1 = -2.772$$ 9. **Solutions to the system:** $$(3.386, 5.772)$$ and $$(-0.886, -2.772)$$ 10. **Minimum value of $$2x^2 - 3x - 7$$:** The quadratic opens upwards (since $$a=2>0$$), so minimum at vertex. Vertex formula for $$x$$: $$x = -\frac{b}{2a} = -\frac{-3}{2 \times 2} = \frac{3}{4} = 0.75$$ 11. **Calculate minimum value:** $$y = 2(0.75)^2 - 3(0.75) - 7 = 2(0.5625) - 2.25 - 7 = 1.125 - 2.25 - 7 = -8.125$$ **Final answers:** - Intersection points: $$(3.386, 5.772)$$ and $$(-0.886, -2.772)$$ - Minimum value of $$2x^2 - 3x - 7$$ is $$-8.125$$ at $$x=0.75$$.