This is why one has to do verification after solving an equation (unless he can guarantee that every step of the solution process is an equivalence ). In the case we consider only the negative or only the positive sqrt, when we square both sides we dont produce an equivalence. In the case we take both then we have an equivalence but there is no way to know which root belongs to which case unless we substitute the solutions to the original equation.
In this example the equation +sqrt(...)=1-x has two roots in R but the equation -sqrt(...)=1-x has no root in R (and no root in C also). I dont see other way to show the latter result, other than squaring, finding the two roots and then by substitution in the original equation to find out that they dont verify it.