What is an extraneous solution and in what cases do you get one?

How do you know it is extraneous?

I have a teacher asking this for College Algebra


Hi Paul,

Two of the definitions of extraneous I found in a dictionary are "not belonging or proper" and "existing or originating outside or beyond". Extraneous solutions are not solutions at all. They arise from outside the problem, from the method of solution. They are extraneous because they are not solutions of the original problem. This answers your second question. To tell if a "solution" is extraneous you need to go back to the original problem and check to see if it is actually a solution.

One example might be

1/(x-1) = x/(x2-1) Solving this algebrically gives x = 1. But this can't be a solution as both denominators are zero when x is 1. This equation has no solution.

Here is another example.

x = sqrt(2x+3) If you square both sides and solve for x you find x = 3 and x = -1. You can check that x = 3 is a solution but x = -1 is not. Substituting x = -1 into the original equation gives -1 = 1 which is cearly false.

These are two ways that extraneous solutions can arise, division by zero and squaring both sides of an equation. They can arise also because of physical constraints. A standard math/physics problem is

"Stand on a 200 meter high building and, at noon, drop a ball. At what time does it hit the ground?"
If you solve this problem you find two times, one after noon and one before noon. The one before noon is clearly extraneous.


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