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This is really a usage question
::For clarity, apply the same technique to solve for ''x'' in <math>x^2 = y</math> - or, see [[Completing the square]]. [[User:Human|Human]] 15:23, 24 April 2007 (EDT)
::For clarity, apply the same technique to solve for ''x'' in <math>x^2 = y</math> - or, see [[Completing the square]]. [[User:Human|Human]] 15:23, 24 April 2007 (EDT)
== This is really a usage question ==
I think we all understand the point that there are two solutions to the equation
x<sup>2</sup> = 9
so what's at stake here is a question of usage: in mathematics and elsewhere, what is the customary meaning of the phrase ''square root'' (does it refer to both solutions to that equation, or only to the positive solution); what is the customary meaning of the radical sign? Usage questions can't be answered by logical reasoning; they always require finding an authority for the usage.
One ordinary dictionary says:
square root NOUN: A divisor of a quantity that when squared gives the quantity. For example, the square roots of 25 are 5 and -5 because 5 × 5 = 25 and (-5) × (-5) = 25[http://www.bartleby.com/61/26/S0682600.html AHD online].
Another says:
Main Entry: square root
Function: noun
: a factor of a number that when squared gives the number <the square root of 9 is ±3>{http://www.m-w.com/dictionary/square%20root Merriam-Webster online)
Wolfram's ''Mathworld'' site, which I've found to be very good, has a long article on [http://mathworld.wolfram.com/SquareRoot.html square root], which opens by saying
:A square root, also called a radical or surd, of x is a number such that r<sup>2</sup> = x. The function is therefore the inverse function of f(x) = x<sup>2</sup> for x ≥ 0.
It goes on to say that
:Note that any positive real number has two square roots, one positive and one negative. For example, the square roots of 9 are -3 and +3.
It then introduces the phrase "principal square root:"
:Any nonnegative real number x has a unique nonnegative square root r; this is called the ''principal square root'' and is written r = x<sup>1/2</sup> or r = √x For example, the principal square root of 9 is +3, while the other square root of 9 is -3. In common usage, unless otherwise specified, "the" square root is generally taken to mean the principal square root.
So this source seems to be saying that the ''phrase" square root includes both values, i.e. there are two "square roots," but that the notation x<sup>1/2</sup> and r = √x refer ''only'' to the principal or positive-valued square root. Thus, according to this source, one could write:
:The two square roots of 9 are ±3.
They don't actually say this but they do use parallel language later, when they say:
:The two square roots of -9 are ±√-9 = ±3''i''
[[User:Dpbsmith|Dpbsmith]] 16:19, 24 April 2007 (EDT)