We use the principle root whenever the radical is within a function, because as most of us know, a function can have only one output value for every one input value. So whenever you are taking derivatives or integrals or merely changing the independent value within a function, the root used will always be the principle (or positive) root.

Whenever we are not dealing with a function relating a independent variable to a dependent one, then both the principle and negative roots must be considered.

Now everything is crystal clear to me. These are the reasons why we have that "convention". I just didn't make the connection before looking into it.

I hope that this clears this up for everyone like me who might not have grasped exactly why we have the "convention" that we do.

]]>But of course, if you have the expression x^2 = 4, you can take the square root of both sides but you must consider the postive and negative square root. x = +- √4 note! If √4 represented the positive AND negative square root of 4, then the +- signs would be uneccessary and redundant.

]]>Thanks again. Oh, I never accept what others say as fact, just another observation, but convention of society says to be polite.

]]>But seriously, never take what anyone says at face value unless you agree with it.

]]>I was not trying to argue with you on this point, I really wasn't sure why the other root was being ignored. Thanks for your insight.

]]>x² = y

±√x² = ±√y

±|x| = ±√y

x = ±√y

There are two solutions to the square root of a non-zero number. For a positive real number, the two square roots are the principle square root and the negative square root. For negative real numbers, the concept of imaginary and complex numbers has been developed to provide a mathematical framework to deal with the results.

However, I will allow that the [b]PRINCIPAL[b] √x² = |x|. I am just wondering

how one is to determine which root is needed for a given problem when it is not stated

explicitly in the problem. Do we always assume that a problem wants the principle root unless

it is specified that both roots are needed?

]]>We say √2^2 is 2, since by convention, the expression asked for the positive square root. If you ask for √(x^2), we want the positive square root, which will only be positive x. Not negative x. Its permissable to move a squared positive constant to the outside of the radical, but a squared variable could either have been a positive or negative number. Either way, its square would be positive, and the square root of a positive number is always positive, thus √x^2 = |x|

]]>Isn't this the reason that the quadratic equation places the ± before the discriminant?

]]>Ricky, isn't the problem really that there are so many instances in which we use the identity √x² = x and it winds up working for our purposes anyway.

If √x² = x works, then so will √x² = |x|. Only one of them is correct however...

Technically, doesn't √x^2 = x and-x?

I think you meant the piece-wise function:

√x^2 = x if x > 0

√x^2 = -x if x ≤ 0

In which case, you'd be right.

]]>√x² ≠ x, this is never valid, whether dealing with limits or not.

√x² = |x|

]]>