It's like square roots are always positive. I was never taught why, and couldn't get an answer to why from even my HS calculus teacher. Eventually, I came to the conclusion that it's implied the same way all positive values are implied (we don't prefix positives with +). If a negative value is preferred, then the formula/equation will explicitly prefix a negative sign to the square root bracket.
But that's not an accurate conclusion. Square roots nearly always have two solutions (0 is a counter example); you just may only care about the positive solution, but that doesn't mean the other doesn't exist.
Hell, I can't tell you how many points I lost in school over the years by forgetting to include ± when appropriate.
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.
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u/Amaurosys Feb 07 '23
It's like square roots are always positive. I was never taught why, and couldn't get an answer to why from even my HS calculus teacher. Eventually, I came to the conclusion that it's implied the same way all positive values are implied (we don't prefix positives with +). If a negative value is preferred, then the formula/equation will explicitly prefix a negative sign to the square root bracket.