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?


  • 5
    $\begingroup$ You do see the term "extraneous solution" sometimes. $\endgroup$ Mar 3 '19 at 13:29
  • $\begingroup$ I saw this term couple of times but it appears to be confusing to me. $\endgroup$
    – Ahmed
    Mar 3 '19 at 13:35
  • $\begingroup$ Why don't you like the first sentence in your example? $\endgroup$
    – Jasper
    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.

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I'll add:

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).

That immediately tells me that the equation can never be true. For some, this might be obvious, but for the fraction of students who are more visual, the graph really makes the point.

  • 4
    $\begingroup$ I ask my students to use implication arrows correctly, and choose between a conditional or biconditional implication between each pair of lines. This forces them to think about which steps can create extraneous solutions. $\endgroup$ Mar 3 '19 at 14:38
  • $\begingroup$ Nice idea, thanks. As I mentioned, the most common extraneous solutions that I see are the result of squaring. The other one is when multiplying both sides of an equation to cancel out a denominator. Any other obvious one you can tell me? $\endgroup$ Mar 3 '19 at 14:49
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    $\begingroup$ Those are the most common ones for sure. If they are applying a function to both sides (such as exponentiating both sides), I like them to point out that this is either biconditional because the function is injective, or merely conditional because the function is not injective. For the record, I am doing this in the context of teaching algebra to future middle school teachers, and they need to understand this stuff better than their students. But any non-injective function gives more examples (applying cosine to both sides for instance). $\endgroup$ Mar 3 '19 at 15:04
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    $\begingroup$ @StevenGubkin At first I read "future middle school students" and I thought wow, those must be some really advanced elementary school students to talking about algebra and bijectivity/injectivity... $\endgroup$
    – Quintec
    Mar 3 '19 at 20:51
  • 1
    $\begingroup$ @Quintec - these terms might be pretty advanced for high school level, but the concept itself is great. And easy to explain. $\endgroup$ Mar 3 '19 at 21:01

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