The given equation for the profit $y$ in terms of the selling price $x$ of the widget is:

$y = -x^2 + 44x - 122$

This equation represents a quadratic function, and since the coefficient of $x^2$ is negative, the parabola opens downward, meaning it has a maximum point. The maximum profit occurs at the vertex of the parabola.

To find the value of $x$ that maximizes $y$, we use the vertex formula for a quadratic equation $y = ax^2 + bx + c$, which gives the x-coordinate of the vertex as:

$x = -\frac{b}{2a}$

In our case, $a = -1$ and $b = 44$, so:

$x = -\frac{44}{2(-1)} = \frac{44}{2} = 22$

Now, substitute $x = 22$ back into the original equation to find the corresponding maximum profit $y$:

$y = -(22)^2 + 44(22) - 122$ $y = -484 + 968 - 122$ $y = 362$

So, the maximum profit the company can make is $362.

Would you like further clarification or details on this solution?

Related Questions:

1. What happens to the profit if the selling price is lower than 22?
2. How do you interpret the discriminant of this quadratic equation in the context of the company’s profit?
3. What would happen to the maximum profit if the coefficient of $x^2$ was positive?
4. How can we determine the break-even points for this profit equation?
5. How would you solve this problem graphically?

Tip:

In a quadratic equation, a negative coefficient of $x^2$ always indicates a maximum point, useful when optimizing real-world situations like profit maximization.