# Quadratic EquationsIntroduction

Welcome to SkateSum, a small company that produces skateboards. Engineers have been working on a brand new model, the *SquareBoard*, which is finally ready to start production. You’ve been put in charge of finding the optimal resale price for the skateboards – and it turns out that building them is not cheap:

- The tools and machines required to construct skateboards cost $5,000. This is often called a
**fixed cost**. - Every skateboard costs additional $30 worth of of wood, other materials, and salary for the employees. This is often called a
**variable cost**.

In other words, the **cost** of producing *n* skateboards is

**cost** =

The new skateboards are highly anticipated, but if the price is too high, fewer people will actually buy one. We can show this on a chart with the price of a skateboard along the *x*-axis, and the corresponding number of people who want buy one (the **demand**) on the *y*-axis.

Which of these charts makes most sense for the relationship between price and demand?

A higher price means that fewer people want to buy a skateboards, so the graph of the function has to move downwards. After doing some market research, economists came up with the following equation:

**demand** = 2800 – 15 × **price**

For example, if a skateboard costs $80, the demand will be

The **revenue** of our company is the total income we make. It is the number of skateboards sold (the *demand*) times the price of each:

**revenue** = **demand** × **price**

But the number we are more interested in is our **profit**: the revenue we make from selling skateboards, minus the cost of producing them. Can you find an equation that expresses our **profit** in terms of just the **price** of every skateboard?

profit | = | revenue − cost |

= |

Notice that this equation contains **price** as well as . It is called a

**Quadratic Equation**

To work out how to maximise our profit, let’s calculate the profit for a few different prices:

price/$ | 20 | 40 | 60 | 80 | 100 | 120 | 140 | 160 | 180 |

profit/$ | –30k | 17k | 72k | 47k | 10k |

Now we can plot all these points in a coordinate system, and connect them with a line:

You’ll remember that the graph of a **Parabola**

If the

We can maximise our profit by pricing the skateboards at approximately $

In the real world, it can be very difficult for companies to determine a precise equation for their profit – and it is likely to be much more complicated than this example.

Still, quadratic equations appear everywhere in nature, engineering and economics. In this course you will learn different methods for solving quadratic equations and to understand their graphs.