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Elimination of radicals - Confused¿?

Problem Solving — algebra and arithmetic (GMAT Focus Edition)
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Elimination of radicals - Confused¿?

by patheinemann » Wed Sep 05, 2012 9:10 am
Actually the question is pretty straight forward, however upon further examination I somehow got confused on the "well-known" fact that anything you do to one side of an algebraic equation, you must do to the other side. I'll try to elaborate with an example:

Given that √(3b-8) = √(12-b), what is b?

I understand that the approach is to cancel both radicals, and then just proceed with a very simple equation. However in strict theory, wouldn't you have to multiply what you do to one side, to the other side as well?

√(3b-8)^2 = √(12-b) * √(3b-8)

I know I am completely wrong, but it is probably only a matter of having too much of this GMAT stuff.
Thanks to anyone that will kindly respond to my inquiry.
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by Jim@StratusPrep » Wed Sep 05, 2012 10:09 am
You are doing the same thing to each side -> you are multiplying them by themselves, which happen to be equal.

Another way of looking at it is adding 1 is the same as adding 9/9.
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by Ian Stewart » Tue Sep 18, 2012 3:05 am
patheinemann wrote:Actually the question is pretty straight forward, however upon further examination I somehow got confused on the "well-known" fact that anything you do to one side of an algebraic equation, you must do to the other side. I'll try to elaborate with an example:

Given that √(3b-8) = √(12-b), what is b?

I understand that the approach is to cancel both radicals, and then just proceed with a very simple equation. However in strict theory, wouldn't you have to multiply what you do to one side, to the other side as well?

√(3b-8)^2 = √(12-b) * √(3b-8)

I know I am completely wrong, but it is probably only a matter of having too much of this GMAT stuff.
Thanks to anyone that will kindly respond to my inquiry.
If you're given a simple equation like x=y, that means that x and y are exactly the same number. If x and y are the same, then their squares must be the same, so x^2 = y^2 must be true.

So in your equation, if √(3b-8) = √(12-b), then √(3b-8) and √(12-b) are the same number, and their squares must be the same, so 3b - 8 = 12 - b.

What you suggested doing is also correct (and I understand why you've asked your question, since when you square the left side, that is the same as multiplying it by √(3b-8)). You can certainly multiply both sides by √(3b-8) if it's useful, but the problem here is that it makes the right side a bit of a mess. In general, with any equation, you can do almost anything you like provided you do it to both sides, so you have an infinite number of options. Part of the art of algebra is determining which of those options will actually lead you closer to the answer you're looking for, and which will just make things more complicated.

And a bit of a technical point: you aren't "cancelling radicals" here. What you're doing is squaring both sides of the equation. I mention that because in a different kind of equation, if you think you can cancel radicals, that might lead to an error. For example, if you see an equation like √a + √b = √c, you cannot "cancel radicals" in any way; it certainly is not true that a + b = c.
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