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The given profit function is:

\[
P(x) = -0.75x^2 + 6x - 9
\]

To determine when there will be profit, we need to find when \(P(x) > 0\). This means we need to solve the inequality:

\[
-0.75x^2 + 6x - 9 > 0
\]

First, let's solve the equation:

\[
-0.75x^2 + 6x - 9 = 0
\]

This will help us identify the critical points where the profit is zero (break-even points). After that, we can determine the intervals where the function is positive (profit exists).

Let's solve this quadratic equation.The solutions to the equation \(-0.75x^2 + 6x - 9 = 0\) are \(x = 2\) and \(x = 6\).

These are the break-even points where the profit is zero. To determine when there is a profit, we need to check the sign of the profit function in the intervals:

1. \(x < 2\)
2. \(2 < x < 6\)
3. \(x > 6\)

By analyzing the quadratic function, we can conclude:

- The function opens downwards (since the coefficient of \(x^2\) is negative).
- The profit is positive between the break-even points, i.e., for \(2 < x < 6\).

Therefore, there will be profit when \(x\) (the number of units) is between 2 and 6.

Would you like a more detailed explanation or further analysis?

Here are 5 related questions:
1. How do you interpret a profit function in terms of economics?
2. What is the significance of the discriminant in quadratic equations?
3. How does the vertex of a quadratic function help in analyzing profit?
4. What are the steps to analyze when a quadratic function is positive or negative?
5. How does the shape of a parabola help in determining intervals of profit?

**Tip:** Always plot the graph of quadratic functions to better understand the intervals where they are positive or negative!

The profit function for selling x units of a certain product is given by P(x) = -0.75x^2 + 6x - 9. For what number of units will there be profit?

Find out for which number of units there will be a profit using the profit function P(x) = -0.75x^2 + 6x - 9. Explore concepts such as quadratic equations, critical points, and profit intervals. Suitable for students in Grades 9-12.

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