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Any suggestions how to solve the following equation:

$|2- \sqrt{n^2+4n} + n| ≥ \frac{1}{10}$

Thank you in advance.

cherry8.8vanilla
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1 Answers1

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Hint: observe that $$ \left|2-\sqrt{n^2+4n}+n\right|\ge\frac1{10}|\Longleftrightarrow\\ \left(2-\sqrt{n^2+4n}+n\ge\frac1{10}\right)\vee\left(2-\sqrt{n^2+4n}+n\le-\frac1{10}\right) $$

Joe
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  • If n is a natural number do you still have to observe: $(2-\sqrt{n^2+4n}+n\le-\frac1{10})$? – cherry8.8vanilla Jun 04 '14 at 09:09
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    Note that $-\sqrt{n^2+4n}+n=n\left(1-\sqrt{1+\frac4n}\right)=n\frac{-\frac4{n}}{1+\sqrt{1+\frac4n}}=-\frac4{1+\sqrt{1+\frac4n}}\in[-2,-1]$ – Joe Jun 04 '14 at 09:37