https://file.aiquickdraw.com/mathbot/file/1727467131181f45sye7r.jpg

The equation provided in the image is:

\[
P(n) = -0.01n^2 + 8n - 700
\]

Where:
- $ n $ is the number of items a retailer puts up for sale each month.
- $ P(n) $ represents the store's monthly profit in dollars.

The question asks: If the store generated \$500 in profit last month, how many items did they put up for sale? We are also instructed to round the answer to the nearest whole item and provide both answers if more than one solution exists.

Thus, we need to solve the equation:

\[
500 = -0.01n^2 + 8n - 700
\]

I'll solve this quadratic equation to find the possible values of $ n $.The solutions to the equation are:

\[
n = 200 \quad \text{or} \quad n = 600
\]

Thus, the retailer could have put up either **200 items** or **600 items** for sale to generate a profit of \$500.

Would you like a deeper explanation, or do you have any further questions?

Here are 5 related questions to explore:

1. How would the profit change if the retailer put up 300 items for sale?
2. What is the maximum profit the store can achieve based on the given profit function?
3. How does the profit function behave for very high or very low numbers of items?
4. If the store wants to achieve a profit of \$1000, how many items should they put up for sale?
5. Can the profit ever be negative? If so, for what values of $n$?

**Tip:** When solving quadratic equations, always consider the possibility of two distinct solutions.

The equation P(n) = -0.01n^2 + 8n - 700 takes n, the number of items that a retailer puts up for sale in their online store each month, and gives P, the store's monthly profit (in dollars). If the store generated $500 in profit last month, how many items did they put up for sale? Round your answer to the nearest whole item. If there is more than one possible answer, give both.

Learn how to solve a quadratic profit equation using algebraic methods. This example problem asks to find the number of items sold by solving the equation P(n) = -0.01n^2 + 8n - 700, where P is the profit. A detailed step-by-step solution shows how to solve for two possible answers.

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