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When solving for the function $x^2=4$ we take the sqaure root of both sides $\sqrt{x^2}=\sqrt{4}$ then you get $x=\pm 2$, obviously this is because square both $2$ and $-2$ will get you $4$. My teacher said that when you simplify ($\sqrt{4}=$) alone the answer is simply $2$, He says that the this is also evident when you graph it and you get only one side of the graph (not the negative value -)

I asked my teacher today who wrote $\sqrt{x^2}=\pm\sqrt{4}$, my response to that was that the square root implies the $\pm$ and by adding it your essentially pulling that out of thin air, I understand that if your simply writing the answer as $\pm{2}$ the $\pm$ makes sense but should you have to include it when you leave it in radical form ($\sqrt{}$), as the square root should imply the ($\pm$)

Blue
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Hadi Beydoun
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    Search this site—this question has been answered already :) The short version: $\sqrt{x^2}$ is equal to $|x|$, not $x$. – Greg Martin Sep 14 '22 at 01:41
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    There is no number $x$ such that $\sqrt{x^2}=-\sqrt4.$ The left side is always positive and the right side is always negative. So it's true that the $\pm$ in the second paragraph is a mistake, but for exactly the opposite reason that you give. Your reason is incorrect. You shouldn't write $\sqrt{x^2}=\pm2$ either; the $-$ sign is not possible in that equation. – David K Sep 14 '22 at 02:06
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    The equation $\sqrt{x^2}=\pm\sqrt4$ says the exact same thing as $\sqrt{x^2}=\pm2,$ because $\sqrt4=2$ and $\sqrt4\neq-2.$ The negative sign in the $\pm$ is a mistake in both cases. The solution $x=\pm2$ is a different thing, because it has $x$ (which can be positive or negative) on the left instead of $\sqrt{x^2},$ which can only be positive. – David K Sep 14 '22 at 02:10
  • Also see Square roots -- positive and negative. – David K Sep 14 '22 at 02:35
  • $x^2=4 > 0 \Leftrightarrow \sqrt{x^2}=\sqrt{4} >0 \Leftrightarrow |x|=2 > 0 \Leftrightarrow$ $x=2 \lor x=-2$, but it is not equivalent to $x=2$. – Ivan Kaznacheyeu Sep 14 '22 at 08:17

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Your teacher is right in the first paragraph and wrong in the second. The square root function always means the nonnegative root. $\sqrt{4} = 2$. The equation $x^2 = 4$ has two roots, $2$ and $-2$, which you can write as $\pm \sqrt{4}$.

Ethan Bolker
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