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What is the $\sqrt{x^2}$? Is it just $x$ or $\pm x$? When is it $\pm x$? So if I had $\sqrt {\sin^2(x)}$ is it then $\sin(x)$?

I feel like its just $x$ as $x^2$ is always positive. Thoughts?

Thanks

frog1944
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  • $|x|{}{}{}{}{}$ – Wojowu May 12 '16 at 07:01
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    "its just $x$ as $x^2$ is always positive": but what if $x<0$ ? –  May 12 '16 at 07:02
  • It can be both. But for some positive number problems ( not always diophantine) we only take positive. Like finding the side of a square given the area. – N.S.JOHN May 12 '16 at 07:02
  • So @YvesDaoust it is $\pm x$? – frog1944 May 12 '16 at 07:04
  • There are two definitions. If you take the square root as "a number that has the given square", than obviously $x^2=(-x)^2$ and $\sqrt{x^2}=\pm x$. The other definition states that the (main branch of the) square root is a positive function, then $\sqrt{x^2}=|x|$. (Notice that $|x|=\pm x$, depending on the sign of $x$.) –  May 12 '16 at 07:16
  • Ok, but if I was trying to solve for $x$ I'd need to take it as the $\pm x$? – frog1944 May 12 '16 at 07:21
  • This has been asked and answered many times on this site. See my answer here for example. – Greg Martin May 12 '16 at 08:08

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Maybe an example can be interesting here. Take $x=-3$. Then $$\sqrt{(-3)^2} = \sqrt{9} =3$$ because a square root of something must be positive by definition. Hence you see that the answer is not $x=-3$ but $3$ which is equal to $-x$. You can do the same with every negative number and will see that $$\sqrt{x^2}=-x$$ if $x$ is negative. Of course $$\sqrt{x^2}=x$$ if $x$ is positive. In conclusion $$\sqrt{x^2}=|x|.$$

C. Dubussy
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