Why Do Radical & Rational Solutions Have Extraneous Solutions?

Matthew Ko

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I'm a junior and I am working on a pre-calculus assignment.

One of the questions are phrased as the following:

Explain why the solutions to (a) radical equations and (b) rational equations must be checked to see if they are extraneous.

I could postulate as far as the fact that they both have inverses which aren't functions; I believe it has to do something with that, but I'm not sure. Could anyone help?

Thank you!
 
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Dr.Peterson

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Explain why the solutions to (a) radical equations and (b) rational equations must be checked to see if they are extraneous.

I could postulate as far as the fact that they both have inverses which aren't functions; I believe it has to do something with that, but I'm not sure. Could anyone help?
There are several ways you could answer this. You have something of the right idea (though you didn't say what "they both" are -- you can't mean the equations themselves, since an equation is not a function).

One thing I consider important is that equations don't have extraneous solutions; solution methods do! That is, sometimes you can solve an equation in two different ways, and only one of them can produce extraneous solutions. So you should be focusing not on the equation itself, but on what you do to it in solving. So, what is it that you do in each case, and why might that produce extraneous solutions? (Your thought about inverses may come in here!)
 

pka

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I'm a junior and I am working on a pre-calculus assignment.
One of the questions are phrased as the following:
Explain why the solutions to (a) radical equations and (b) rational equations must be checked to see if they are extraneous.
I could postulate as far as the fact that they both have inverses which aren't functions; I believe it has to do something with that, but I'm not sure.
Read this brief link.
 

Matthew Ko

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There are several ways you could answer this. You have something of the right idea (though you didn't say what "they both" are -- you can't mean the equations themselves, since an equation is not a function).

One thing I consider important is that equations don't have extraneous solutions; solution methods do! That is, sometimes you can solve an equation in two different ways, and only one of them can produce extraneous solutions. So you should be focusing not on the equation itself, but on what you do to it in solving. So, what is it that you do in each case, and why might that produce extraneous solutions? (Your thought about inverses may come in here!)
Thanks for your answer! But could you help me? I still can't quite connect the dots.
 

Subhotosh Khan

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Thanks for your answer! But could you help me? I still can't quite connect the dots.
Did you read the link provided in response #3?

If you did - please tell us "exactly" where you are getting lost?

I think your questioned is answered in first few sentences (similar to response #2)!
 

Dr.Peterson

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Thanks for your answer! But could you help me? I still can't quite connect the dots.
I was, in effect, asking you to state what you are doing in solving each kind of equation, and then consider how that might lead to an extraneous solution. It's different in each case. If doing that for yourself doesn't reveal the answer, tell us what you've done to look for it.

The link to Wikipedia does state the answer in each case, if you don't want to go through the whole thinking process yourself; and very likely your textbook said some of the same things. (Note that the article is organized around what you do, not what kind of equation it is, just as I suggested.)

You may also find this blog post of mine helpful, looking at answers to students questions along the same line.
 

Matthew Ko

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Thanks for all your help! I now figured it out!
 
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