Understanding Quadratic Equations and Profit

Quadratic equations are very useful in math, especially when solving real-life problems like figuring out how to make the most money. In this blog post, we’ll look at how these equations work in business and share some examples to help everyone understand.

What Are Quadratic Equations?

A quadratic equation usually looks like this:

$$y = ax^2 + bx + c$$

Here’s what that means:

• $a$, $b$, and $c$ are numbers we use in the equation.
• $x$ is the variable, which can stand for things like how many products we sell.
• $y$ usually represents profit or money made.

When $a$ is a positive number, the graph looks like a "U," and the lowest point shows the minimum. When $a$ is negative, the graph flips upside down, showing the highest point, which is what we care about in business when we want to maximize profit.

Profit as Part of a Quadratic Equation

Let’s make this clearer with an example. Imagine a bakery that sells a special pastry:

• Each pastry is priced at $p$ pounds.
• The number of pastries sold is $x$.
• The profit can often be shown in a quadratic way. For example, the profit function could look like this:

$$P(x) = -2x^2 + 20x - 30$$

In this function:

• $-2x^2$ means that as more pastries are made, the profit on each one decreases because of factors like costs.
• $20x$ shows that profits go up when more pastries are sold, at least up to a point.
• $-30$ represents fixed costs the bakery has to pay.

How to Find the Maximum Profit

To find out where the maximum profit happens, we need to find the vertex of this curve. The vertex gives us the highest profit point.

The x-coordinate of the vertex can be found using this formula:

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

In our case, $a = -2$ and $b = 20$. Plugging in those numbers gives us:

$$x = -\frac{20}{2 \times -2} = 5$$

So, the best way for the bakery to make money is to produce and sell 5 pastries.

Calculating Maximum Profit

Now, let’s find out how much money the bakery makes when they sell 5 pastries by putting $x = 5$ back into the profit function:

$$P(5) = -2(5)^2 + 20(5) - 30$$

Let’s break this down step-by-step:

1. Calculate $5^2 = 25$, then $-2(25) = -50$.
2. Calculate $20(5) = 100$.
3. Now put it back into the profit function:

$$P(5) = -50 + 100 - 30 = 20.$$

So, the most profit the bakery can make is $20.

Conclusion

Using our bakery example, we can see how quadratic equations help us understand and solve problems like finding maximum profit. By figuring out the profit function and its highest point with the vertex formula, businesses can make smart choices. Knowing these concepts not only helps with math but also gives useful insights into economics and running a business.