Math CentralQuandaries & Queries


Question from Lenval, a teacher:

Why do quadratic equations arising from practical problems often give physically impossible solutions such as negative distances, negative times or, say, a polygon with a negative number of sides? For example: "A river in flood carries a tree t-squared - 23t metres downstream in t seconds. How long does it take for the tree to be carried 50 metres downstream?" This gives t=25 and t=-2. Please explain the second 'solution', which is not physically possible.


It is a question of the domain of the function. The practical problem requires that the domain be the non-negative real numbers. The method of solution (probably the quadratic formula) solves the equation for the domain of all real numbers. That's ok, you just have to exclude ones that are not in the domain you care about; they are not solutions to your problem.

Hope this helps.

Lenval wrote back

Thanks to Victoria West for the quick response, but I do have a follow-up query it that's allowed. She replies to my question about physically-impossible 'solutions' to quadratic equations (such as negative distances and negative times) by saying that they are not solutions at all since the function that generates them only applies over a certain restricted domain, i.e. the non-negative reals. Is this an implicit recognition that mathematicians routinely employ classes of numbers which have no known connection to what we commonly refer to as 'the real world'?



Dear Lenval,

In a way, the answer to your question is yes. In some sense one could look at the solution having been obtained by using methods that allow one to step outside of the physical restrictions temporarily. But, the mathematical model of a physical situation should specify all of the parameters and constraints of interest. One of these is the domain of any function that is described. When a model consists of one or more equations that must be solved to determine a particular value of interest, then any method can be used to generate possibilities. What is required (at the end) is that the solver check that the possible solution(s) generated meet all of the criteria (e.g. constraints) necessary for it to correspond to a physical solution. The checking is actually a very important part of the solution method.

I hope this makes sense.



Perhaps a better way to think of unwanted numbers as "not relevant to the matter at hand." No number is any more real than any other, nor can numbers be classified as useful or useless. Often the unwanted numbers arise from the method of solution. As a very simple example, if I start with x = -2, and I square both sides of the equation, I get the true statement that x2 = 4. But if I tell you only that x2 = 4, there would be no natural way for you to recover the number x that I started with.
Quadratic equations often arise naturally from some equation that describes a given relationship in the world of finance or of science. Suppose that you determine that x is a number that satisfies the equation
x = √(x+6).

Sometimes you can guess the answer; if not, you can obtain the solution by squaring both sides to get
x2 = x + 6.
You recognize this as a quadratic equation whose roots are 3 and -2. To know which root is relevant, you would have to know the original situation. If the problem comes from economics, x = -2 might mean that you have lost money; from physics it might mean an object fell 2 metres. All the algebra tells you is that for the square of a number to equal the number plus six, that number would have to be 3 or -2.


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