6 Quadratic equations (p. 168 - 170)

Learning objectives

After completing this chapter students should be able to:

· Use factorization to solve quadratic equations with one unknown variable.
· Use the quadratic equation solution formula.
· Identify quadratic equations that cannot be solved.
· Set up and solve economic problems that involve quadratic functions.
· Construct a spreadsheet to plot quadratic and higher order polynomial functions.

6.1 Solving quadratic equations

A quadratic equation is one that can be written in the form

ax² + bx + c = 0

where x is an unknown variable and a, b and c are constant parameters with a =/ 0. For example,

6x² + 2.5x + 7 = 0

A quadratic equation that includes terms in both x and x² cannot be rearranged to get a single term in x, so we cannot use the method used to solve linear equations.
There are three possible methods one might try to use to solve for the unknown in a quadratic equation:

(i) by plotting a graph
(ii) by factorization
(iii) using the quadratic ‘formula’

In the next three sections we shall see how each can be used to tackle the following question. If a monopoly can face the linear demand schedule

p = 85 - 2q (1)

at what output will total revenue be 200?

It is not immediately obvious that this question involves a quadratic equation.We first need to use economic analysis to set up the mathematical problem to be solved. By definition we know that total revenue will be

TR = pq (2)

So, substituting the function for p from (1) into (2), we get

TR = (85 - 2q)q = 85q - 2q²

This is a quadratic function that cannot be ‘solved’ as it stands. It just tells us the value of TR for any given output. What the question asks is ‘at what value of q will this function be equal to 200’? The mathematical problem is therefore to solve the quadratic equation

200 = 85q - 2q² (3)

All three solution methods require like terms to be brought together on one side of the equality sign, leaving a zero on the other side. It is also necessary to put the terms in the order given in the above definition of a quadratic equation, i.e. unknown squared (q²), unknown (q), constant Thus (3) above can be rewritten as

2q² - 85q + 200 = 0

It is this quadratic equation that each of the three methods explained in the following sections will be used to solve.
Before we run through these methods, however, you should note that an equation involving terms in x² and a constant, but not x, can usually be solved by a simpler method. For example, suppose that

5x² - 80 = 0

this can be rearranged to give

5x² = 80
x² = 16
x = 4

6.2 Graphical solution

Drawing a graph of a quadratic function can be a long-winded and not very accurate process that involves separately plotting each individual value of the variable within the range that is being considered. It is therefore usually not a very practical method of solving a quadratic equation. The graphical method can be useful, however, not so much for finding an approximate value for the solution, but for explaining why certain quadratic equations do not have a solution whilst others have two solutions. Only a rough sketch diagram is necessary for this purpose.