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This is a really basic question and to be honest I feel ashamed to be asking this when I'm in precalculus and trigonometry right now.

When I was younger, I was taught that $\sqrt{x^2}$ was equal to $\pm x$. However, during this course I've inputted the negative before but was told that it was wrong, and that instead it was only the positive solution, not the negative as well. I tried contacting the professor numerous times but they haven't responded in weeks so I figured I'd come to here. Any help would be greatly appreciated.

Andrei
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rosa
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  • The definition of the surd is the positive root. $\sqrt{9}=3$ is definition, but $(\pm \sqrt{9})^2=3$ – Gareth Ma Apr 15 '20 at 03:33
  • Precisely, its $|x|$ –  Apr 15 '20 at 03:33
  • that is not true, $a^{2}=x^{2}$ has solution $a=\pm x$. A function however cannot have more than one value. $\sqrt{x^{2}}=|x|$ – acat3 Apr 15 '20 at 03:34
  • Thanks so much, haha! Again, I feel really bad asking about this. I'm surprised I made it to precalc without understanding radicals – rosa Apr 15 '20 at 03:35
  • Or this: https://math.stackexchange.com/questions/1448885/square-root-confusion – Gerry Myerson Apr 15 '20 at 03:35
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    @rosa nothing to be ashamed of. In the starting chapter of real analysis by T.Tao one has to prove something like $3 \ne 5$ –  Apr 15 '20 at 03:37
  • To throw in some jargon here, you can talk about the "principal" square root. This simply refers to the positive square root. For example, the principal square root of 4 is 2, but -2 and 2 both square to 4. – morrowmh Apr 15 '20 at 03:43

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You shouldn't be taught that $\sqrt{x^2}=\pm x$. The definition of $\sqrt{x^2}$ is $|x|$. $\sqrt{}$ is a function and can have only one value.

What you learned before, is that there are two solutions to $x^2-a^2$ - namely $x=\pm a$. Those can be written as $x^2-a^2=0 \implies x=\pm\sqrt{a^2}=\pm a$. But $\sqrt{x}$ is a function and only has one value (the positive one if $x\geq 0$, the primitive root otherwise).

Gareth Ma
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  • @CroCo Is the updated version correct? (Last line) – Gareth Ma Apr 15 '20 at 03:47
  • @CroCo You’ve gotten that backwards. The square of a negative real is a positive Real, so the square root of a complex number could never be a negative real. Including complex values lets you take square roots of negative reals. – amd Apr 15 '20 at 08:32