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

Given: $R(x) = 892.70x - 7x^2$ $C(x) = 3x + 19572.28$

Step 1: Set $R(x) = C(x)$

$892.70x - 7x^2 = 3x + 19572.28$

Step 2: Move all terms to one side of the equation

$-7x^2 + 892.70x - 3x - 19572.28 = 0$

Simplify the terms:

$-7x^2 + 889.70x - 19572.28 = 0$

Step 3: Rewrite in standard form

$7x^2 - 889.70x + 19572.28 = 0$

Step 4: Solve the quadratic equation

We can solve this quadratic equation $7x^2 - 889.70x + 19572.28 = 0$ using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where $a = 7$, $b = -889.70$, and $c = 19572.28$.

Calculate $x$:

1. Calculate the discriminant: $b^2 - 4ac = (-889.70)^2 - 4 \cdot 7 \cdot 19572.28$

2. Solve for $x$:

Using these values in the formula, we can find the break-even points.

Let me compute these values for you.The break-even points are approximately:

$x = 98.80, \quad x = 28.30$

These values represent the quantities $x$ at which revenue equals cost.

Would you like a detailed explanation of each step, or do you have further questions?

Here are five related questions that expand on this information:

1. How can we interpret the break-even points in a real-world context?
2. What happens to the profit if we produce quantities greater than the highest break-even point?
3. How would changes in the cost or revenue functions affect the break-even points?
4. Can there be cases where break-even points do not exist in a quadratic function?
5. How can we graphically verify the break-even points?

Tip: Remember that solving quadratic equations often yields two answers, which represent critical points in revenue-cost analyses.