The given equation for the company's profit, $y$, as a function of the selling price of each widget, $x$, is:

$y = -5x^2 + 209x - 1090$

This is a quadratic equation of the form $y = ax^2 + bx + c$, where:

• $a = -5$
• $b = 209$
• $c = -1090$

Finding the Price for Maximum Profit:

For a quadratic equation, the maximum (or minimum) value occurs at the vertex. Since $a < 0$ (i.e., $a = -5$), the parabola opens downwards, meaning there is a maximum profit at the vertex.

The formula for the $x$-coordinate of the vertex is:

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

Substitute the values of $a$ and $b$ into this formula:

$x = \frac{-209}{2(-5)} = \frac{-209}{-10} = 20.9$

Thus, the price per widget that maximizes profit is $20.90.

Conclusion:

The company should sell the widgets for $20.90 to maximize its profit.

Would you like further details or any clarification?

---

Here are 5 related questions to deepen your understanding:

1. What is the maximum profit the company can make?
2. How would the profit change if the selling price is increased beyond $20.90?
3. What happens to the profit if the company decreases the price to $15?
4. How would the result change if the coefficient of $x^2$ were different?
5. Can the company still make a profit if the price is set at $10?

Tip: In any quadratic profit equation, when the coefficient of $x^2$ is negative, the maximum occurs at the vertex!