A company sells widgets. The amount of profit, y, made by the company, is related to the selling price of each widget, x, by the given equation. Using this equation, find out the maximum amount of profit the company can make, to the nearest dollar.
y, equals, minus, x, squared, plus, 75, x, minus, 636
y=−x 
2
 +75x−636

The given profit equation is:

\[
y = -x^2 + 75x - 636
\]

This is a quadratic equation of the form $y = ax^2 + bx + c$, where $a = -1$, $b = 75$, and $c = -636$.

### Finding the maximum profit:
Since the coefficient of $x^2$ is negative ($a = -1$), the parabola opens downwards, meaning there is a maximum point (vertex).

The formula to find the $x$-coordinate of the vertex of a parabola $y = ax^2 + bx + c$ is:

\[
x = \frac{-b}{2a}
\]

Substitute $a = -1$ and $b = 75$:

\[
x = \frac{-75}{2(-1)} = \frac{75}{2} = 37.5
\]

Now that we have the $x$-coordinate of the vertex, we can substitute $x = 37.5$ into the original equation to find the maximum profit:

\[
y = -(37.5)^2 + 75(37.5) - 636
\]

Let's compute this.The maximum profit the company can make is approximately \$770.

Would you like further details or have any questions? Here are some related questions:

1. How do changes in the coefficients of a quadratic equation affect its graph?
2. What is the significance of the vertex in a quadratic function in terms of real-world applications?
3. How do we interpret the maximum or minimum point in a quadratic profit model?
4. Can there be cases where a company doesn't make any profit according to such models?
5. What is the relationship between the vertex and axis of symmetry in parabolas?

**Tip**: The vertex formula $\frac{-b}{2a}$ is essential for finding maximum or minimum values in quadratic functions!

A company sells widgets. The amount of profit, y, made by the company, is related to the selling price of each widget, x, by the given equation. Using this equation, find out the maximum amount of profit the company can make, to the nearest dollar. y = −x^2 + 75x − 636

Learn how to calculate the maximum profit a company can make using the quadratic equation y = −x^2 + 75x − 636. This problem involves key algebraic concepts such as quadratic equations and the vertex formula. Ideal for students in Grades 9-12.

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