Quadratic Equations - Mathematics and Statistics for the Environment - page 97

5 Quadratic Equations

2

ax ++= bx c 0

2

= −± −

bb ac
a

4

x

2

When studying a discipline that does not include mathematics, one thing is certain: The discipline
under study has little or nothing to do with environmental practice.

5.1 QUADRATIC EQUATIONS AND ENVIRONMENTAL PRACTICE
A logical question at this point might be why is the quadratic equation important in environmental
practice? The logical answer is that the quadratic equation is used in environmental practice to find
solutions to problems primarily dealing with length and time determinations. Stated differently:
The quadratic equation is a tool, an important tool that belongs in every environmental practitio-
ner's toolbox.
To the student of mathematics, this explanation might seem somewhat strange. Math students
know, for example, that there will be two solutions to a quadratic equation. In environmental disci-
plines such as environmental engineering, many times only one solution is meaningful. For example,
if we are dealing with a length, a negative solution to the equation may be mathematically possible
but is not the solution we would use. Negative time, obviously, would also pose the same problem.
So what is the point? The point is that we often need to find a solution to certain mathematical
problems. In environmental problems involving the determination of length and time using qua-
dratic equations, we will end up with two answers. In some instances, a positive answer and a nega-
tive answer may result. One of these answers is usable; thus, we would use it. Real engineering is
about modeling situations that occur naturally and using the model to understand what is happening
or maybe to predict what will happen in future. The quadratic equation is often used in modeling
because it is a beautifully simple curve (Bourne, 2013).

5.1.1 K ey t erms
• a is the coefficient of x 2 .
• b is the coefficient of x .
• c is a number in the quadratic equation (not a coefficient of any x term).
• Simple equations are equations in which the unknown appears only in the irst degree.
• Pure quadratic equations are equations in which the unknown appears only in the second
degree.
• Affected quadratic equations are equations containing the irst and second degree of an
unknown.

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