I don't understand how x^2 = 9 where x = +/- 9 translates to a formula intuitively. imagine this question below:
sqrt((d+3)^2). I am assuming the below is equivalent =+(d+3) and -(d+3)
my question is why? I don't understand how D + 3 = negative and positive because the expression itself will provide a positive or negative, but the expression itself should not change.
are we factoring out a negative 1 from the expression d+3 so that we can represent it as both positive and negative?
below is the question i was simplifying that got me thinking about this.
Most of the time $\sqrt{x}$ is meant to be a positive number but as you've noticed algebraically the roots are indistinguishable and there isn't any reason to prefer one to the other. This will be apparent in the complex numbers as well when you can substitute $-i$ for $i$ everywhere and it all works exactly the same because they are again indistinguishable as roots of the polynomial $x^2+1$. These substitutions come up again in Galois Theory where you study the permutations of these indistinguishable roots to derive strong results on the solvability of polynomial equations, among other things.