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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.

Given the profit function P(n) = -0.25n^2 + 2.5n - 2.25, determine where the function is positive and where it is negative.

Determine the intervals where the profit function P(n) = -0.25n^2 + 2.5n - 2.25 is positive or negative. Learn how to solve this quadratic equation using the quadratic formula and analyze its behavior over different intervals. Suitable for high school students in Grades 9-12.

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