The function you provided is:

\[
P(x) = 5000 + 500x - 50x^2
\]

This represents a quadratic function, often used to model profit, revenue, or cost in various contexts, where \( x \) might represent the number of units or time periods, and \( P(x) \) represents the profit, revenue, or cost.

### Key insights about this function:
1. **General Form**: This is a quadratic equation of the form \( ax^2 + bx + c \), where \( a = -50 \), \( b = 500 \), and \( c = 5000 \). Since \( a \) is negative, the parabola opens downward, indicating that this function has a maximum point.
   
2. **Vertex (Maximum Value)**: The maximum or minimum of a quadratic function occurs at the vertex, which can be calculated using the formula:
   \[
   x_{\text{vertex}} = \frac{-b}{2a}
   \]
   Substituting \( a = -50 \) and \( b = 500 \):
   \[
   x_{\text{vertex}} = \frac{-500}{2(-50)} = \frac{-500}{-100} = 5
   \]
   The vertex occurs at \( x = 5 \). This means that at \( x = 5 \), the function reaches its maximum value.

3. **Maximum Value**: To find the maximum value of \( P(x) \), substitute \( x = 5 \) back into the equation:
   \[
   P(5) = 5000 + 500(5) - 50(5)^2
   \]
   \[
   P(5) = 5000 + 2500 - 50(25)
   \]
   \[
   P(5) = 5000 + 2500 - 1250 = 6250
   \]
   So, the maximum value of \( P(x) \) is 6250.

4. **Zeros of the Function**: To find when \( P(x) = 0 \), we can solve the quadratic equation:
   \[
   5000 + 500x - 50x^2 = 0
   \]
   This can be solved using the quadratic formula:
   \[
   x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   \]
   Where \( a = -50 \), \( b = 500 \), and \( c = 5000 \).

Would you like me to proceed with solving the quadratic equation to find the roots?

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**Relative Questions:**
1. What is the meaning of the vertex in practical applications of this equation?
2. How does the sign of \( a \) influence the shape of a quadratic function?
3. What other methods can be used to find the maximum or minimum of a quadratic function?
4. How does the discriminant help in determining the nature of the roots of a quadratic equation?
5. How can this function be used to model real-world business situations?

**Tip:** For any quadratic equation, the vertex formula is key to quickly identifying the maximum or minimum without needing to graph the function.

Explore how to solve the quadratic function P(x)=5000+500x-50x² to find the maximum profit. Learn the steps to calculate the vertex and roots, and understand key concepts like maxima and minima in quadratic equations. Ideal for students in Grades 10-12.

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