1. The problem is to analyze the quadratic function $$P = -42x^2 + 581x - 527$$ which represents profit as a function of the number of hundred basketball conference championship t-shirts sold, denoted by $x$. 2. Since the coefficient of $x^2$ is negative ($-42$), the parabola opens downward, indicating there is a maximum profit point. 3. To find the vertex (maximum profit), use the vertex formula for $x$-coordinate: $$x = -\frac{b}{2a} = -\frac{581}{2 \times (-42)} = \frac{581}{84} \approx 6.9167$$ 4. Substitute $x \approx 6.9167$ back into the profit function to find the maximum profit: $$P = -42(6.9167)^2 + 581(6.9167) - 527$$ Calculate step-by-step: $$6.9167^2 \approx 47.85$$ $$-42 \times 47.85 = -2009.7$$ $$581 \times 6.9167 = 4019.6$$ So, $$P = -2009.7 + 4019.6 - 527 = 1482.9$$ 5. Interpretation: The maximum profit is approximately 1483 dollars when about 691.67 (since $x$ is in hundreds) t-shirts are sold. 6. To find the roots (break-even points), solve $-42x^2 + 581x - 527 = 0$ using the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-581 \pm \sqrt{581^2 - 4(-42)(-527)}}{2(-42)}$$ Calculate discriminant: $$581^2 = 337561$$ $$4 \times 42 \times 527 = 88536$$ $$\sqrt{337561 - 88536} = \sqrt{249025} = 499.02$$ Calculate roots: $$x_1 = \frac{-581 + 499.02}{-84} = \frac{-81.98}{-84} = 0.976$$ $$x_2 = \frac{-581 - 499.02}{-84} = \frac{-1080.02}{-84} = 12.86$$ 7. Interpretation: Profit is zero when approximately 98 and 1286 t-shirts are sold (in hundreds). Final answers: - Maximum profit: approximately 1483 dollars at $x \approx 6.92$ - Break-even points: $x \approx 0.98$ and $x \approx 12.86$