Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

$$\pm x = \pm 3$$

Thanks!

share|improve this question
    
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 at 9:57

4 Answers 4

up vote 4 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 $x=\pm3$?

share|improve this answer

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.

share|improve this answer

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

share|improve this answer

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 :-)

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.