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As the title says, why when you take a square root of both sides of the equation do you add $\pm$ only to the side which is a number, as opposed to an unknown?

For example:

$$x^2 = 9 \implies x = \pm3$$

Why isn't it like below?

$$\pm x = \pm 3$$


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I have seen some authors write, for instance, $\pm (x^2+1) = 2 \mp 3$ meaning $+(x^2+1) = 2-3$ and $-(x^2+1) = 2+3$. So to avoid confusion I would not write $\pm x = \pm 3$ because it could seem like the pluses belong together and the minuses belong together, resulting in only one solution, namely $x = 3$. – Improve May 31 '14 at 9:57
up vote 5 down vote accepted

Let's write $\pm x=\pm 3$.
We have $4$ cases:
1. $x=3$
2. $x=-3$
3. $-x=3\implies x=-3$
4. $-x=-3\implies x=3$

Case 1 and case 4 are the same and case 2 and case 3 are also same. So why not just write $x=\pm3$?

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There are a couple of ways to look at this. I think about it as solving a puzzle (not a very complicated one). The equation says, "I'm thinking of a number that when you square it, you get 9. What could the number be?" The answer to that is that $x =3$ or $x = -3$ - those are the only answers, which is exactly what the equation $x = \pm 3$ means.

You can also think about it that putting the $\pm$ on both sides is redundant - the equation $\pm x = \pm 3$ contains 4 different possibilities: $x = 3$, $x = -3$, $-x = 3$, or $-x = -3$. But those possibilities only lead to two solutions: $x=3$ and $x-3$. So the other two equations are redundant.

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Because $x=-y$ is equivalent to $-x=y$.

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If it confuses you (and I can see how it might), don't use the $\pm$ symbol at all. Just write "$x=3$ or $x=-3$". That's perfectly fine. It means exactly the same as $x = \pm3$, and it's clearer. The only problem is that it's bit longer. You can switch back to the shorthand $\pm$ form later, if your arm gets tired or you run short of ink :-)

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