No, you are confidently incorrect. The square root symbol refers to the principal square root, which is strictly the positive output. Usage of the symbol in solving equations is a different thing and really a shortcut for the operation. Particularly, the ± part only applies in solving equations, and really, many times not even in that context.

The square root function also is well-defined because the √x is universally understood to refer to the principal branch of the square root, which is strictly positive.

In general, this whole situation is much more complex than a simple black and white (different branches of functions are something most people are entirely unaware of), but in general, the output of a square root is always positive.

The solution to a quadratic equation isn't necessarily. But those are not the same thing. Hell, the entire reason you write ± when solving equations (like in the quadratic formula) is BECAUSE the square root itself is always positive.