Yahoo Answers: Answers and Comments for Please Help with the proof of Arithmetic and Geometric mean theorem? [Mathematics]
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From Anonymous
enUS
Tue, 01 Feb 2011 10:35:47 +0000
3
Yahoo Answers: Answers and Comments for Please Help with the proof of Arithmetic and Geometric mean theorem? [Mathematics]
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From Pauley Morph: It's used to perform a sort of trick. Addi...
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Tue, 01 Feb 2011 18:49:21 +0000
It's used to perform a sort of trick. Adding the same number to both sides of an inequality preserves the inequality, so there is nothing wrong with adding the 4ab to both sides. The nice this is, it converts (ab)^2 on the left side into (a+b)^2 and that turns out to be very useful later on in the proof. If your asking how somebody could come up with the proof listed there, I can't answer that. Often, once someone has come up with a proof of something, they will try to find a better proof. Once you know something is true, you can spend all of your time being clever. It's sort of like machismo.

From robby: If you have an inequality, you can always add ...
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Tue, 01 Feb 2011 18:39:30 +0000
If you have an inequality, you can always add any term to both sides and keep the inequality. In this case, 4ab is what is necessary to be added to change (ab)^2 into (a+b)^2.