1. **State the problem:** The band wants to earn 500 by selling posters. The number of posters sold depends on the price per poster $p$ and is given by the expression $-3.5p + 85$. 2. **Write the revenue formula:** Revenue $R$ is price per poster times number of posters sold: $$R = p \times (-3.5p + 85)$$ 3. **Set revenue equal to 500 and form the quadratic equation:** $$p(-3.5p + 85) = 500$$ $$-3.5p^2 + 85p = 500$$ 4. **Rewrite the equation in standard form:** $$-3.5p^2 + 85p - 500 = 0$$ 5. **Multiply entire equation by $-1$ to simplify:** $$\cancel{-}3.5p^2 + \cancel{-}85p + \cancel{-}500 = 0 \Rightarrow 3.5p^2 - 85p + 500 = 0$$ 6. **Use the quadratic formula:** $$p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=3.5$, $b=-85$, $c=500$. 7. **Calculate the discriminant:** $$b^2 - 4ac = (-85)^2 - 4(3.5)(500) = 7225 - 7000 = 225$$ 8. **Calculate the roots:** $$p = \frac{85 \pm \sqrt{225}}{2 \times 3.5} = \frac{85 \pm 15}{7}$$ 9. **Find the two possible prices:** $$p_1 = \frac{85 + 15}{7} = \frac{100}{7} \approx 14.29$$ $$p_2 = \frac{85 - 15}{7} = \frac{70}{7} = 10$$ 10. **Interpretation:** Both prices yield revenue of 500, but the question asks for the highest price per poster. **Final answer:** To the nearest dollar, the highest price per poster is **14**.