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Quadratic EquationsIntroduction

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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 units.

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=revenuecost
=

Notice that this equation contains price as well as price2. It is called a Quadratic Equation, named after the Latin word “quadratus” for square.

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

price/$20406080100120140160180
profit/$–30k17kkkkk72k47k10k

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 linear function is always a straight line. But as you saw above, the graph of a quadratic function is curved. It even has a specific name: a Parabola.

If the price is 0, our profit is negative, because we’re just giving away expensive skateboards for free. As the price increases, our profits rise, too. However, if the skateboards become too expensive, people no longer want to buy them and our profit falls again.

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

In the real word, 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.