is there a way to prevent false roots?

Question

Is there a systematic way of preventing false roots when squaring a root equation? The testing of the roots is quite tedious in some problems.

My first thought was absolute values in some form

Answer 1

Let $A$ and $B$ denote mathematical expressions which evaluate to real numbers. There are two cases:

If you have an equation of the form $\sqrt{A} = B$, it is equivalent to $A = B^2, B \geq 0$.

If you have an equation of the form $\sqrt{A} = \sqrt{B}$, it is equivalent to $A = B \geq 0$.

So basically you have to transform one equation into an equation plus an inequality to have an equivalence transform.