1. **Problem:** A football player punts a ball. The path of the ball is modeled by the equation $$h = -0.004d^2 + d + 25$$ where $d$ is the horizontal distance in feet and $h$ is the height in feet. We need to find how far from the player the ball will land, i.e., when $h=0$. 2. **Formula and rules:** To find where the ball lands, set the height $h=0$ and solve the quadratic equation: $$0 = -0.004d^2 + d + 25$$ We will use the quadratic formula: $$d = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a = -0.004$, $b = 1$, and $c = 25$. 3. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 1^2 - 4(-0.004)(25) = 1 + 0.4 = 1.4$$ 4. **Apply the quadratic formula:** $$d = \frac{-1 \pm \sqrt{1.4}}{2(-0.004)}$$ 5. **Simplify denominator:** $$2(-0.004) = -0.008$$ 6. **Calculate roots:** $$d_1 = \frac{-1 + \sqrt{1.4}}{-0.008}, \quad d_2 = \frac{-1 - \sqrt{1.4}}{-0.008}$$ 7. **Evaluate square root:** $$\sqrt{1.4} \approx 1.1832$$ 8. **Calculate each root:** $$d_1 = \frac{-1 + 1.1832}{-0.008} = \frac{0.1832}{-0.008} = -22.9$$ $$d_2 = \frac{-1 - 1.1832}{-0.008} = \frac{-2.1832}{-0.008} = 272.9$$ 9. **Interpretation:** Distance cannot be negative, so the ball lands approximately at $$d = 272.9$$ feet from the player. **Final answer:** The ball will land about **273 feet** from the football player.