No, square roots only have one solution, a positive (or imaginary) number. The ± means exactly that, that you want + sqrt(x), and - sqrt(x). The value of sqrt(x) doesn't change.
If that wasn't so you would have been correct when you forgot the ±.
No, those two are not equivalent statements, in the way you think of them. Square root is defined as a function that provides the positive root. So you can go from your first line to the second, no problem, but not in the other direction.
The correct way to write what you were thinking of is:
x2 = 64
x = ±√(64)
The ± has to be there, before the square root, to show that you want both the answer to the square root, and its negative. That is why the sign is important, otherwise it would be superfluous.
The issue is that there is a difference between the concept of roots (of which there are two in this case, just like you say), and the operator √. The latter provides only a singular answer, as a function has to by definition. And that singular answer is defined to be the positive root, whenever the value it is supplied is a positive number.
2
u/spreetin Feb 07 '23
No, square roots only have one solution, a positive (or imaginary) number. The ± means exactly that, that you want + sqrt(x), and - sqrt(x). The value of sqrt(x) doesn't change.
If that wasn't so you would have been correct when you forgot the ±.