1. **Problem statement:** A sporting goods store sells 90 ski jackets at 200 each. Each 10 decrease in price results in 5 more jackets sold. Find the number of jackets and price to get revenue 17600. 2. **Define variables:** Let $x$ be the number of 10 decreases in price. 3. **Express price and quantity:** Price per jacket: $200 - 10x$ Number of jackets sold: $90 + 5x$ 4. **Revenue formula:** $$R = \text{price} \times \text{quantity} = (200 - 10x)(90 + 5x)$$ 5. **Set revenue to 17600:** $$(200 - 10x)(90 + 5x) = 17600$$ 6. **Expand:** $$200 \times 90 + 200 \times 5x - 10x \times 90 - 10x \times 5x = 17600$$ $$18000 + 1000x - 900x - 50x^2 = 17600$$ 7. **Simplify:** $$18000 + 100x - 50x^2 = 17600$$ 8. **Bring all terms to one side:** $$-50x^2 + 100x + 18000 - 17600 = 0$$ $$-50x^2 + 100x + 400 = 0$$ 9. **Divide entire equation by -50:** $$\cancel{-50}x^2 + \cancel{100}x + \cancel{400} = 0 \Rightarrow x^2 - 2x - 8 = 0$$ 10. **Solve quadratic:** $$x = \frac{2 \pm \sqrt{(-2)^2 - 4 \times 1 \times (-8)}}{2} = \frac{2 \pm \sqrt{4 + 32}}{2} = \frac{2 \pm \sqrt{36}}{2}$$ $$x = \frac{2 \pm 6}{2}$$ 11. **Two solutions:** $$x = \frac{2 + 6}{2} = 4 \quad \text{or} \quad x = \frac{2 - 6}{2} = -2$$ 12. **Reject negative $x$ (price decrease cannot be negative):** $$x = 4$$ 13. **Calculate price and quantity:** Price: $200 - 10 \times 4 = 200 - 40 = 160$ Quantity: $90 + 5 \times 4 = 90 + 20 = 110$ 14. **Check revenue:** $$160 \times 110 = 17600$$ **Final answer:** Number of jackets = 110 Price per jacket = 160