To find the **break-even point**, we need to determine when the profit \( P(x) \) equals zero. The given profit function is:

\[
P(x) = -0.75x^2 + 4x - 2.8
\]

At the break-even point, \( P(x) = 0 \). Thus, we set the equation to zero and solve for \( x \):

\[
-0.75x^2 + 4x - 2.8 = 0
\]

We will use the quadratic formula to solve this, which is:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Where:
- \( a = -0.75 \)
- \( b = 4 \)
- \( c = -2.8 \)

Let's calculate the break-even points.The break-even points are approximately:

\[
x_1 = 0.83 \quad \text{and} \quad x_2 = 4.50
\]

This means that the company will break even when producing and selling between approximately 0.83 and 4.50 units of the product.

Would you like further details or have any questions?

Here are five related questions:
1. How would changes in the cost structure affect the break-even points?
2. What do the solutions represent in terms of profit and loss?
3. How does the quadratic nature of the profit function affect decision-making?
4. What is the maximum profit, and at what point does it occur?
5. How would external factors like price changes alter this break-even calculation?

**Tip**: To further understand break-even analysis, consider also calculating the **marginal cost** and **marginal revenue** to optimize profit.

Learn how to solve the break-even analysis problem with the quadratic equation P(x) = -0.75x^2 + 4x - 2.8. This solution walks you through using the quadratic formula to find the break-even points, ideal for students in Grades 9-12.

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