So, I'm thinking, if a problem includes the square root function, but no plus or minus, you assume the primary root, however, if you create a square root when solving a problem you need to be aware of the negative root in the context of the question....

Please read the other responses to this post because they are good and will give you different perspectives.

\(\displaystyle \sqrt{a} \ge 0\) by definition if we are dealing with real numbers.

So, we do not assume that the square root is the primary root (meaning a non-negative number), we know that it is a non-negative number by definition.

However, if the square root of a is positive, we know that the square of the square root's additive inverse is also a. In notation,

\(\displaystyle \sqrt{a} * \sqrt{a} = a = (-\ \sqrt{a}) * (-\ \sqrt{a}).\)

So if we have an equation like

\(\displaystyle x^2 = c \ge 0\),

we have no way to know from the equation itself whether

\(\displaystyle x = \sqrt{c} \text { or } x = - \ \sqrt{c}.\)

It may be that either solution works or that only one of the two works, but that must be decided based on information not contained in the equation itself.

Uncertainty about which answer applies does not mean that there is any uncertainty about what the surd means. To clarify where the uncertainty lies we say

\(\displaystyle x = \pm \sqrt{c}.\)

The sign of the surd is certain. It is the sign of x that is uncertain.