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The given profit function is:

\[
P(x) = -2x^2 + 60x - 250
\]

We are asked to determine two things:
1. The factors that can be used to find the zeroes of the function (i.e., the points where \( P(x) = 0 \)).
2. The number of units the company should make and sell to earn a profit.

### Step 1: Finding the factors
To find the factors, we need to solve for \(x\) when \(P(x) = 0\).

\[
-2x^2 + 60x - 250 = 0
\]

First, divide the whole equation by \(-2\) to simplify:

\[
x^2 - 30x + 125 = 0
\]

Next, we solve this quadratic equation using the quadratic formula:

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

For the equation \(x^2 - 30x + 125 = 0\), we have:
- \(a = 1\)
- \(b = -30\)
- \(c = 125\)

Substitute these values into the quadratic formula:

\[
x = \frac{-(-30) \pm \sqrt{(-30)^2 - 4(1)(125)}}{2(1)}
\]

\[
x = \frac{30 \pm \sqrt{900 - 500}}{2}
\]

\[
x = \frac{30 \pm \sqrt{400}}{2}
\]

\[
x = \frac{30 \pm 20}{2}
\]

Thus, the two solutions are:

\[
x = \frac{30 + 20}{2} = 25 \quad \text{and} \quad x = \frac{30 - 20}{2} = 5
\]

So the factors are \( (x - 25) \) and \( (x - 5) \).

### Step 2: Finding the interval for profit
For the company to make a profit, it needs to produce units between the zeroes of the function. The zeroes are \(x = 5\) and \(x = 25\), so the company should make and sell between 5 and 25 units to earn a profit.

### Final Answers:
- **Factors**: \( (x - 25) \) and \( (x - 5) \)
- **Units to make and sell**: Between 5 and 25 units

Would you like further details or have any questions?

Here are 5 related questions to expand on this:
1. How do you determine whether a quadratic function opens upward or downward?
2. What is the significance of the vertex in the profit function?
3. Can you explain how to complete the square for this quadratic function?
4. How does solving the quadratic equation help in business decision-making?
5. What is the relationship between the roots of the quadratic function and the break-even points?

**Tip**: The discriminant \(\Delta = b^2 - 4ac\) of a quadratic equation tells you the nature of the roots (real or complex).

A profit function for a product made by a company is given as P(x) = -2x^2 + 60x - 250. What are the factors that can be used to determine the zeroes of the function? How many units of the product should the company make and sell to earn a profit?

Explore how to solve a quadratic profit function using the quadratic formula. The problem focuses on finding the factors and the zeroes of P(x) = -2x^2 + 60x - 250 to determine the profitable units to produce and sell. Suitable for students in Grades 8-10.

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