That is a pretty common misconception.

x*x = a, this implies x = sqrt(a) or -sqrt(a)

**But the square root itself always yields a positive number; it is by definition.** So, sqrt (1) = 1 and NOT -1.

The problem with the -1 = 1 "proof" is that sqrt(-1)*sqrt(-1) DOES NOT EQUAL sqrt (-1*-1). Why? Because sqrt(a)*sqrt(b) = sqrt(ab) is true only if at least one of a and b is non-negative.

It is relatively advanced Maths, although still school level, but I absolutely loved this one when I was back in school. Complex numbers can be complex