# The terminology for an excluded solution

I am not a native math teacher. I have a question related to a terminology when solving an algebraic equation.

Assume that we are solving some complicated equation like $$x^{3}-\sqrt{1-x^{2}}-\frac{1}{x+1}=2$$. After doing a bunch of algebraic manipulation, we come up with some finite possible solutions: $$x=1$$, $$x=2$$,.... Now, because I see that $$x=1$$ is not a solution (for example, by inserting into the equation to check). Then I might say that " $$x=1$$ does not satisfy the equation, so it is not a solution.". However I don't like this sentence. I would prefer to use " $$x=1$$ is excluded by not satisfying the equation". Is it ok with this?

Thanks.

• You do see the term "extraneous solution" sometimes. Mar 3 '19 at 13:29
• I saw this term couple of times but it appears to be confusing to me. Mar 3 '19 at 13:35
• Why don't you like the first sentence in your example? Mar 11 '19 at 14:17

As noted in a comment, the word for this is "extraneous solution".

I explain it to high school students this way:

When you are trying to remove the square root, i.e. by squaring both sides of an equation, you risk introducing extraneous solutions, e.g. $$1 = -1$$ (not true), whereas after squaring, $$1 = 1$$ is indeed true.

I'm sure it can arise from other manipulations, but this one seems most common.

I find that both for students and myself, graphing almost always adds to an understanding of the answer involved. I can analyze your equation and see that for the fact that $$x=1$$ is the highest $$x$$ can go (higher, and you have the square root of a negative number, same for -1, $$x$$ cannot be lower).