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: $R(x) = 125.40x - 2x^2$ $C(x) = 13x + 1489.44$

Setting $R(x) = C(x)$:

$125.40x - 2x^2 = 13x + 1489.44$

Rearrange to form a standard quadratic equation:

$-2x^2 + 125.40x - 13x - 1489.44 = 0$ $-2x^2 + 112.40x - 1489.44 = 0$

Dividing through by $-2$ for simplicity:

$x^2 - 56.20x + 744.72 = 0$

Now, we solve this quadratic equation using the quadratic formula:

$x = \frac{-(-56.20) \pm \sqrt{(-56.20)^2 - 4 \cdot 1 \cdot 744.72}}{2 \cdot 1}$ $x = \frac{56.20 \pm \sqrt{3160.44 - 2978.88}}{2}$ $x = \frac{56.20 \pm \sqrt{181.56}}{2}$ $x = \frac{56.20 \pm 13.47}{2}$

Calculating the two possible values for $x$:

1. $x = \frac{56.20 + 13.47}{2} = 34.835$
2. $x = \frac{56.20 - 13.47}{2} = 21.365$

So, the break-even points are approximately: $x \approx 34.84, 21.37$

Would you like a step-by-step explanation for solving quadratics, or do you have any questions?

Related Questions

1. How is the break-even point relevant in business decisions?
2. What happens if the revenue function is less than the cost function for all values of $x$?
3. Can the break-even points change if fixed costs change?
4. How do we determine the maximum profit given a revenue and cost function?
5. What are other methods to find break-even points without solving equations?

Tip: Always double-check your quadratic equation setup; small errors can lead to incorrect roots.