https://file.aiquickdraw.com/mathbot/file/17267763537135y3im28i.jpg

Let's analyze the problem.

### (a) Factor the right side of the equation:

We are given:
\[
R = 780x - x^2
\]
This is a quadratic expression. To factor it, we take out the common factor, which is \( x \):

\[
R = x(780 - x)
\]

Thus, the factored form of the equation is:
\[
R = x(780 - x)
\]

### (b) What is the expression for the price of the item?

Revenue \( R \) is the product of the price per unit and the number of units sold \( x \). The expression for the revenue in factored form is:
\[
R = x(780 - x)
\]
From this, we see that the price per item is \( 780 - x \).

So, the price of the item is:
\[
\text{Price per item} = 780 - x
\]

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Do you need further clarification on any of these steps?

Here are some related questions to deepen your understanding:
1. What does the value \( x \) represent in the context of this problem?
2. What would the graph of the revenue function look like?
3. What happens to the revenue when \( x = 780 \)?
4. How would you find the maximum revenue using this function?
5. What is the significance of factoring in this case?

**Tip:** Whenever you see a quadratic expression like \( ax^2 + bx \), factoring can often help reveal useful information about the behavior of the function!

Revenue R from the sale of x units of a product is found by multiplying the price by the number of items sold. Factor the right side of R = 780x - x^2 and find the expression for the price of the item.

This problem demonstrates how to factor the quadratic expression R = 780x - x^2 to find an expression for price. Suitable for high school students, it involves algebraic concepts such as factoring and revenue functions.

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