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The square root is always positive, but you can plug it into the quadratic formula to get the two possible values.
Nope. To clarify, square roots are the opposite of squaring.
Now ask yourself:
What is 10² ?
What is (-10)² ?
If you get the same answer, then they are both the roots of the answer. +10 and -10 then gets together called ±10
There's no reason to bring the quadratic formula into this. Square roots can be negative, but when talking about the square root it's normally assumed to be the principal square root, which is the positive one.