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

Given: $R(x) = 472.80x - 4x^2$ $C(x) = 12x + 7464.60$

The break-even point occurs where $R(x) = C(x)$. So, we set up the equation: $472.80x - 4x^2 = 12x + 7464.60$

1. Rearrange the equation: $472.80x - 4x^2 - 12x - 7464.60 = 0$ Simplify terms: $-4x^2 + 460.80x - 7464.60 = 0$

2. Rewrite as a standard quadratic equation: $4x^2 - 460.80x + 7464.60 = 0$

3. Solve for $x$ using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where $a = 4$, $b = -460.80$, and $c = 7464.60$.

4. Calculate the discriminant: $b^2 - 4ac = (-460.80)^2 - 4 \times 4 \times 7464.60$

5. Find the values of $x$:

Let's perform the calculations.The break-even points are at $x = 95.7$ and $x = 19.5$.

Would you like further details or have any questions?

Here are some related questions to explore:

1. How do we interpret the meaning of multiple break-even points in this context?
2. What happens to the profit at values of $x$ between the break-even points?
3. How would the break-even points change if the cost function had a different slope?
4. How can we verify the solution by substituting back into the original equations?
5. How does the discriminant affect the type and number of solutions?

Tip: Remember that break-even points indicate where revenue exactly matches costs, meaning no profit or loss.