1. **Problem statement:** Find the length of the line segment AB where the line $y = x + 4$ intersects the parabola $y = \frac{1}{2}x^2$. 2. **Set the equations equal to find intersection points:** $$x + 4 = \frac{1}{2}x^2$$ Multiply both sides by 2 to clear the fraction: $$2x + 8 = x^2$$ Rewrite as a quadratic equation: $$x^2 - 2x - 8 = 0$$ 3. **Factor the quadratic:** $$(x - 4)(x + 2) = 0$$ So the solutions are: $$x = 4 \quad \text{or} \quad x = -2$$ 4. **Find corresponding y-values:** For $x=4$: $$y = 4 + 4 = 8$$ For $x=-2$: $$y = -2 + 4 = 2$$ 5. **Coordinates of points A and B:** $$A = (-2, 2), \quad B = (4, 8)$$ 6. **Calculate length of segment AB using distance formula:** $$AB = \sqrt{(4 - (-2))^2 + (8 - 2)^2} = \sqrt{6^2 + 6^2} = \sqrt{36 + 36} = \sqrt{72}$$ 7. **Simplify the square root:** $$\sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2}$$ **Final answer:** $$AB = 6\sqrt{2}$$