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?

---

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.