Square Root, +/- sign?

[Tobias Funke]

Statdad really did answer this like 5 posts in, but Dr. Mirrage, you seem to be getting caught up with writing $$\sqrt{x^2}=\pm x$$. But this is interpreted as saying that both $$x$$ and $$-x$$ are (nonnegative) square roots of $$x^2$$, which simply isn't true unless x=0. So $$\sqrt{x^2}=\pm x$$ is wrong, while $$\sqrt{x^2}=|x|$$ is correct.

And since $$\sqrt{9}=3$$, the clearest way to write this is $$|x|=3$$, as Statdad said. Maybe going right to $$x=\pm 3$$ from $$x^2=9$$ is confusing you because it makes it seem like something is happening with the signs on the right side, when the right side is straightforward.

By the way, this is a very good question. You'll find that high school math glosses over many key points either to save time or avoid opening up a can of worms.

 

[statdad]

Statdad really did answer this like 5 posts in, but Dr. Mirrage, you seem to be getting caught up with writing $$\sqrt{x^2}=\pm x$$. But this is interpreted as saying that both $$x$$ and $$-x$$ are (nonnegative) square roots of $$x^2$$, which simply isn't true unless x=0. So $$\sqrt{x^2}=\pm x$$ is wrong, while $$\sqrt{x^2}=|x|$$ is correct.

And since $$\sqrt{9}=3$$, the clearest way to write this is $$|x|=3$$, as Statdad said. Maybe going right to $$x=\pm 3$$ from $$x^2=9$$ is confusing you because it makes it seem like something is happening with the signs on the right side, when the right side is straightforward.

By the way, this is a very good question. You'll find that high school math glosses over many key points either to save time or avoid opening up a can of worms.

A (final?) point:
since $$\sqrt{x^2}$$ as written is the square root function , it must return a single value: writing $$\sqrt{x^2} = \pm x$$ indicates that this is not the case, whereas $$\sqrt{x^2} = |x|$$ does show a single value as the result.

 

[Dr. Mirrage]

Ok this is all making sense. But I'm not sure that I understand why $\sqrt{9}$ is defined as just 3. Because I was always told and it also seems to make sense to me that $\sqrt{9}$ is plus or minus 3. Also if we go with absolute values then don't we still have a second solution that is -|x| ?
I'll try not to carry this on much longer, but if someone could just explain the above then I think I would be satisfied.

 

[Tobias Funke]

Ok this is all making sense. But I'm not sure that I understand why $\sqrt{9}$ is defined as just 3. Because I was always told and it also seems to make sense to me that $\sqrt{9}$ is plus or minus 3.

The $\sqrt$ symbol returns the nonnegative root by definition, in order for it to define a single valued function (and actually, by every definition I've seen, multivalued "functions" aren't really functions.)

What you're thinking of is different. You're thinking of the set of solutions of the equation $x^2=9$, which is of course $\{-3,3\}$. It's basically just a notational thing.

Also if we go with absolute values then don't we still have a second solution that is -|x| ?

I'll try to explain how I view the solution and what to make of the absolute value. If we have the equation $x^2=9$, since both sides are positive we can take the positive square root of each and keep the equality. This does really depend on existence and uniqueness of square roots, but it should be obvious. This is where the right side is straightforward. $\sqrt{9}=3$.

For the left side, the positive root is not $x$ because, for example, $\sqrt{(-3)^2}=\sqrt{9}=3.$. However, it is true that $\sqrt{x^2}=|x|$, so we get the new equality $|x|=3$. It's not too hard to see that this operation is reversible, so our original equation and $|x|=3$ have the same solution set, and the latter equation has solution set $\{-3,3\}$.

Hope that helps a little. Keep asking until you fully understand- that's what we're here for!

 

[Dr. Mirrage]

Thanks, I think I get it now, I had just never heard that the sqrt sign was defined as the positive root before. I know a few people on this thread said it too but I didn't really consider it and it makes sense now that I think about it.

So, then the only question I still have is when should I use the plus/minus sign? because I still see people using it.. or is it just sloppy notation that should be avoided?

 

[Hurkyl]

So, then the only question I still have is when should I use the plus/minus sign?

When it makes things easier -- when it makes it easier for you to (correctly) solve a problem, when it makes it easier for you to explain something, and so forth.

If you don't find it makes things easier, then don't use it.