Square root and principal square root confusion A few months ago I asked a question about the $\pm$ symbol because I was confused about it... I still carry the same confusion (which really bugs me) but I think the real confusion has to do with the square root and principal square root. I hope I can finally grasp the concept with the following two questions...
Question 1:
$\sqrt{12x^2} = \sqrt{4\cdot3x^2} = 2|x|\sqrt{3}$
Why do we only have to keep in mind that $x$ could be negative, yet we just factorize $\sqrt{12}$ to $\sqrt{3\cdot4}$ and put the 2 right in front of the radical sign?
$\sqrt{12}$ could be factorized as $\sqrt{-3 \cdot -4}$ as well.
Question 2:
I've been told to take the square root of both sides in the following equation, but the square root doesn't have its own symbol right? Only the principal root does... So if you'd use the radical sign you're using the principal root and therefore you're missing out on a solution in the following scenario:
$(x + 8)² = 1 \iff \sqrt{(x + 8)^2} = \sqrt1 \iff x + 8 = 1 \iff x = -7$
Does that mean that whenever we want to take the square root of something (not the principal) we just use a $\pm$ symbol instead? For example:
$(x + 8)^2 = 1 \iff x + 8 = \pm 1 \iff x = -8\pm1$

For question 2, you could disambiguate the situation by factoring. Remember that the plus-or-minus symbol is just a convenience, and, like most conveniences, it involves some loss of something, clarity in this case.
Nothing is free, and the cost of convenience is loss of clarity, or safety (as in trying to cross the track before the oncoming train passes), or something else.
