Introduction

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 variablefixed 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 downwardsupwards.

After doing some market research, economists came up with the following equation:

demand = 2800 – 15 × price - (A)

For example, if a skateboard costs $80, the demand will be 16001000 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 - (B)

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? We know that:

profit = revenue - cost - (C)

Notice that the equation (C) contains price as well as price2 when we substitute equation (A) and (B). It is called a Quadratic Equation, named after the Latin word “quadratus” for squaresquare root.

Now, to work out how to maximise our profit, let’s calculate the profit for a few different prices and plot all these points in a coordinate system, and connect them with a curve:

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 $108 ± 10109.

Just like in multiple scenarios in the the real world, situations arise which give rise to us forming mathematical expressions which have a general form containing variable2 as one of its terms.

Let's take another example: For instance, a community centre is to built with a podium having an area of 150 square metres with its length being one metre more than twice its breadth.

What should be the length and breadth of the hall?

Suppose the breadth of the hall is x metres. Then, its length should be (2x + 1) metres.

Now, area of the hall = (2x + 1)× x m2 = 2x2+xm2

So, 2x2+x=150 which can be further written as: 2x2+x−150=0

So, the breadthlength of the hall should satisfy the equation 2x2+x−150=0 which is another quadratic equation.

One common type of quadratic polynomial comes in the form: ax2+bx+c, a ≠ 0.

When we equate this polynomial to zero, we get a quadratic equation.

Solving of quadratic equations, in general form, is often credited to ancient Indian mathematicians. In fact, Brahmagupta (C.E. 598 – 665) gave an explicit formula to solve a quadratic equation of the form ax2+bx=c. Later, Sridharacharya (C.E. 1025) derived a formula, now known as the quadratic formula, for solving a quadratic equation by the method of completing the square.

An Arab mathematician Al-Khwarizmi (about C.E. 800) also studied quadratic equations of different types. Abraham bar Hiyya Ha-Nasi, in his book ‘Liber embadorum’ published in Europe in C.E. 1145 gave complete solutions of different quadratic equations.

In this chapter, we will study quadratic equations, and various ways of finding their roots. We will also see some applications of quadratic equations in daily life situations.