Isn't it you only consider to get the positive and negative root if it's like this:
[itex]x^2=25[/itex]
Then that's when you say that [itex]x=\pm5[/itex]

Is there any law or postulate that clears this situation??
Please correct me if I am wrong and if my teacher is actually correct.

And if I am correct, how can I explain it to her so that she'll get it right away? I'm not that good at explaining anyway.

You need to distinguish between "root of a number" and the "square-root *function*". Every piece of software I know of treats the square-root _function_ (sqrt or sqr or ....) as a single POSITIVE value (provided that you start with a positive value whose root you want). However, there are two *roots* to the equation x^2 = a (a > 0), namely: x = sqrt(a) and x = -sqrt(a). '

But if we have [itex]\sqrt{x} ; x = 25[/itex] Then the answer is [itex]\pm 5[/itex]? So that it depends on whether we are taking the root of a number or variable? Or is it only if we have a value squared, and we put the squareroot sign that we use [itex]\pm[/itex]?

Staff: Mentor

The way I know it, those two are not equal. But my teacher insists on saying that whenever there is a square root, we should always recognize also the negative root.

This is a good question, frozonecom. Apparently there are regional differences, so the correct answer depends on the country you are in (or the country where your teacher studied).

The answers you received in this forum indicate the square root symbol to be taken as denoting the positive square root. I, however, studied along with the same nomenclature as your teacher, and I require recognition of both + and - answers for the explicit square root symbol.

Another area where there are regional differences is in expressions such as 2/5x
On some forums this is staunchly defended as being equal to 0.4/x
On this forum, I'm pleased to see it is just as strongly regarded to be 0.4x

This is a good question, frozonecom. Apparently there are regional differences, so the correct answer depends on the country you are in (or the country where your teacher studied).

The answers you received in this forum indicate the square root symbol to be taken as denoting the positive square root. I, however, studied along with the same nomenclature as your teacher, and I require recognition of both + and - answers for the explicit square root symbol.

Another area where there are regional differences is in expressions such as 2/5x
On some forums this is staunchly defended as being equal to 0.4/x
On this forum, I'm pleased to see it is just as strongly regarded to be 0.4x

With clarity being crucial to mathematics, I endorse only the latter.

If clarity is crucial to mathematics, then it would seem that you ought to endorse never typing 2/5x, and always only typing either 2/(5x) or (2/5)x.

Regarding the original post, if [itex] x - \sqrt{2} = 0 [/itex], then [itex] x = \sqrt{2} [/itex], and that's IT. Surely this is true REGARDLESS of whether you conventionally think of [itex] \sqrt{2} [/itex] as denoting only +1.414..., or whether you think of it as something that could equally well denote either +1.414.. or -1.414...

Either way, the answer is [itex] x = \sqrt{2} [/itex]. An answer of [itex] -\sqrt{2} [/itex] wouldn't make sense, because as dextercioby has already pointed out, plugging this value back into your original equation, you see that it is not a solution:[tex] -\sqrt{2} - \sqrt{2} \neq 0 [/tex]

Now, for those of you who apparently adopt the convention that the radical sign can denote either the positive or the negative root, then I suppose you could argue that the first radical is meant to denote -1.414..., and the second radical is meant to denote +1.414..., in which case you do get 0. However, in that case, you have the same symbol used to represent two different numbers in the same equation, and nothing whatsoever to indicate which one is which. This seems like a really bad notational practice to me.

The convention in which the square root sign denotes only the positive root makes a lot more sense to me.

then the solution will be the angles where sin is positive which is on QI and QII.

Questions like this also need statement regarding restrictions for x (like -π/2 ≤ x ≤ π/2 or x in ℝ). If there was a restriction of -π/2 ≤ x ≤ π/2, for instance, then you shouldn't look for the angle in Q II at all.

Staff: Mentor

While you are a student, you calculate answers the way your lecturer wants you to. :)

If you are unsure, then I think in an exam it is worth playing safe. Work it out assuming the radical means the positive only. Then, write, "if we are required to also consider the negative of the square root then ...." and provide a further solution.

You are likely to lose few marks if your lecturer doesn't entertain the negative alternative, and you should get full marks if your lecturer expects that you'll examine both outcomes.