Find the break-even point(s) for the revenue and cost functions below. Separate multiple answers with a comma.

R(x)=892.70x−7x2C(x)=3x+19572.28


To find the break-even point(s) for the given revenue and cost functions, we need to set the revenue function \( R(x) \) equal to the cost function \( C(x) \) and solve for \( x \).

### Given functions:
\[
R(x) = 892.70x - 7x^2
\]
\[
C(x) = 3x + 19572.28
\]

### Step 1: Set \( R(x) = C(x) \)
\[
892.70x - 7x^2 = 3x + 19572.28
\]

### Step 2: Rearrange the equation to set it equal to zero:
\[
-7x^2 + 892.70x - 3x - 19572.28 = 0
\]
\[
-7x^2 + 889.70x - 19572.28 = 0
\]

### Step 3: Solve the quadratic equation
This is a standard quadratic equation of the form:
\[
ax^2 + bx + c = 0
\]
Where:
\[
a = -7, \quad b = 889.70, \quad c = -19572.28
\]

Use the quadratic formula to solve for \( x \):
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Substituting the values:
\[
x = \frac{-889.70 \pm \sqrt{(889.70)^2 - 4(-7)(-19572.28)}}{2(-7)}
\]

### Step 4: Calculate the discriminant
\[
\text{Discriminant} = 889.70^2 - 4(-7)(-19572.28)
\]
\[
\text{Discriminant} = 791564.09 - 54803.84
\]
\[
\text{Discriminant} = 736760.25
\]

### Step 5: Compute the solutions
\[
x = \frac{-889.70 \pm \sqrt{736760.25}}{-14}
\]
\[
x = \frac{-889.70 \pm 858.46}{-14}
\]

Now, calculate the two possible values of \( x \):

1. \( x_1 = \frac{-889.70 + 858.46}{-14} = \frac{-31.24}{-14} = 2.23 \)
2. \( x_2 = \frac{-889.70 - 858.46}{-14} = \frac{-1748.16}{-14} = 124.87 \)

### Conclusion
The break-even points are:
\[
x = 2.23, \quad x = 124.87
\]

Would you like further details or have any questions?

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Here are 8 related questions for further practice:

1. What is the revenue at the break-even points?
2. Calculate the cost at the break-even points.
3. What are the conditions for no break-even points in a quadratic revenue function?
4. How does the discriminant in the quadratic formula affect the number of break-even points?
5. What is the significance of each term in the quadratic revenue function?
6. What is the minimum or maximum value of the revenue function \( R(x) \)?
7. How does changing the coefficient of \( x^2 \) in the revenue function affect the break-even points?
8. If the cost function had a higher fixed cost, how would that affect the break-even points?

**Tip:** When solving quadratic equations, always check the discriminant to determine if there are real, repeated, or complex solutions.

Learn how to find break-even points for revenue and cost functions using quadratic equations. Step-by-step solution for solving revenue and cost functions, including detailed calculations and interpretations. Suitable for high school students in Grades 10-12.

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