Quadratic equations are very helpful for businesses when it comes to making money. We can think of money earned as revenue, which we often write as $R$. This revenue can be represented by a quadratic function that relies on pricing choices and how many items are sold.

Example: Ticket Sales

Let’s take a movie theater as an example. The price of a ticket affects how many tickets the theater sells. We can express this relationship with the equation:

$R(p) = -5p^2 + 100p$

In this equation, $p$ is the ticket price. Because the $p^2$ part (the coefficient) is negative, this function opens downwards. This tells us there is a highest point for revenue.

Finding Maximum Revenue

To find the best price to earn the most money, we look for the vertex of this parabolic shape. We can find the price $p$ using this formula:

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

For our equation, $a = -5$ and $b = 100$:

$p = -\frac{100}{2 \times -5} = 10$

So, if the ticket price is set at $10, we can calculate the maximum revenue as follows:

$R(10) = -5(10)^2 + 100(10) = 500$

By understanding quadratic equations, businesses can make smart pricing choices. This helps them earn more money and improve their revenue!