To solve the problem, we need to analyze the profit function $P(n) = -0.25n^2 + 2.5n - 2.25$ and determine where the function is positive or negative.

Step 1: Find the roots of the quadratic equation

We set $P(n) = 0$ to find the points where the function crosses the x-axis (the roots): $-0.25n^2 + 2.5n - 2.25 = 0$ Dividing through by $-0.25$ to simplify: $n^2 - 10n + 9 = 0$ Now we solve the quadratic equation using the quadratic formula: $n = \frac{-(-10) \pm \sqrt{(-10)^2 - 4(1)(9)}}{2(1)}$ $n = \frac{10 \pm \sqrt{100 - 36}}{2}$ $n = \frac{10 \pm \sqrt{64}}{2}$ $n = \frac{10 \pm 8}{2}$ Thus, the two solutions are: $n = \frac{10 + 8}{2} = 9 \quad \text{and} \quad n = \frac{10 - 8}{2} = 1$ So the roots are $n = 1$ and $n = 9$.

Step 2: Analyze the sign of the function

The quadratic function $P(n)$ opens downwards because the coefficient of $n^2$ is negative. The function will be positive between the roots $n = 1$ and $n = 9$, and negative outside this interval.

• For $n < 1$, the function is negative.
• For $1 < n < 9$, the function is positive.
• For $n > 9$, the function is negative.

Conclusion

The function is positive for $1 < n < 9$ and negative otherwise.

Thus, the correct answer is: d) The function is positive for $1 < n < 9$, and negative for $n < 1$ and $n > 9$.

Would you like more details on this solution, or have any questions?

Follow-up questions:

1. What is the importance of finding the roots of a quadratic function?
2. Can a quadratic function have more than two real roots?
3. How does the sign of the leading coefficient affect the shape of a quadratic function?
4. What method can be used to factor quadratics without using the quadratic formula?
5. How can the discriminant help in determining the nature of the roots?

Tip: When solving quadratic equations, simplifying the equation by dividing by common factors (if possible) can make the process easier and reduce errors.