C(x)=3x^(2)-10x+200
R(x)=-2x^(2)+100x+50
find x

To find the value of \(x\) where cost and revenue are equal (the break-even points), we need to solve the equation where \( R(x) = C(x) \).

### Step 1: Set the two functions equal to each other
\[
-2x^2 + 100x + 50 = 3x^2 - 10x + 200
\]

### Step 2: Rearrange the equation
Move all terms to one side of the equation:
\[
-2x^2 + 100x + 50 - 3x^2 + 10x - 200 = 0
\]
Simplifying:
\[
-5x^2 + 110x - 150 = 0
\]

### Step 3: Solve the quadratic equation
Now we solve the quadratic equation \( -5x^2 + 110x - 150 = 0 \). We can either use the quadratic formula or factor it. Let's use the quadratic formula, where:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
For the equation \( -5x^2 + 110x - 150 = 0 \), we have:
- \( a = -5 \)
- \( b = 110 \)
- \( c = -150 \)

Substituting these values into the quadratic formula:

\[
x = \frac{-110 \pm \sqrt{110^2 - 4(-5)(-150)}}{2(-5)}
\]
\[
x = \frac{-110 \pm \sqrt{12100 - 3000}}{-10}
\]
\[
x = \frac{-110 \pm \sqrt{9100}}{-10}
\]
\[
x = \frac{-110 \pm 95.39}{-10}
\]

### Step 4: Find the two solutions
Now, calculate the two possible values for \(x\):
1. \( x_1 = \frac{-110 + 95.39}{-10} = \frac{-14.61}{-10} = 1.461 \)
2. \( x_2 = \frac{-110 - 95.39}{-10} = \frac{-205.39}{-10} = 20.539 \)

Thus, the break-even points occur at:
\[
x_1 \approx 1.46 \quad \text{and} \quad x_2 \approx 20.54
\]

These are the two values of \(x\) where cost equals revenue, meaning the company breaks even.

Would you like further details or explanation on these results?

### 5 Related Questions:
1. What is the total profit at the break-even points?
2. How does the profit change between the two break-even points?
3. What production level maximizes profit?
4. At what level does revenue exceed costs the most?
5. How can you interpret negative profits outside of the break-even points?

#### Tip:
For quadratic functions like these, analyzing the discriminant (\(b^2 - 4ac\)) can quickly tell you if real solutions exist.

Learn how to find the value of x where the cost function C(x) equals the revenue function R(x). Detailed steps and calculations provided for solving the quadratic equation formed by equating C(x) and R(x). Suitable for high school students and anyone interested in quadratic equations and business applications.

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