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The problem provided is a quadratic word problem, and the equation given is:

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

This is a quadratic equation in the standard form $ y = ax^2 + bx + c $, where:
- $ a = -1 $
- $ b = 75 $
- $ c = -636 $

### Finding the Maximum Profit

Since the quadratic term $ -x^2 $ has a negative coefficient, the parabola opens downward, meaning it has a maximum point at the vertex. The formula to find the x-coordinate of the vertex for a quadratic equation $ ax^2 + bx + c $ is:

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

Substituting the values of $ a $ and $ b $:

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

Now that we have $ x = 37.5 $, we can substitute this value back into the original equation to find the corresponding maximum profit $ y $:

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

Let's calculate this value to determine the maximum profit.The maximum profit the company can make is approximately **$770**, rounded to the nearest dollar.

Would you like more details or further clarification?

Here are 5 related questions to consider:
1. How do you find the vertex of a quadratic equation graphically?
2. What does it mean for a parabola to "open downwards" in terms of its maximum or minimum value?
3. How can you identify if a quadratic equation has a maximum or minimum value just by looking at the equation?
4. What is the significance of the coefficients $a$, $b$, and $c$ in a quadratic equation?
5. How does the discriminant of a quadratic equation relate to its solutions?

**Tip:** The x-value of the vertex in a quadratic equation represents the point where the maximum or minimum occurs, depending on the direction of the parabola.

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

Find out how to calculate the maximum profit for a company selling widgets using the quadratic equation y = -x^2 + 75x - 636. Learn about algebraic concepts such as quadratic equations, vertex calculation, and profit maximization. Suitable for high school students in Grades 9-12.

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