it's not a real one.
First \(-1^2 = -1\) and it's not 1
Second, if you mean (-1)2 then it's all good, but the last part. If you solve the equation x2=1 ......
Method 1:
\(\color{red}x^2=1\)
\(\color{red}\sqrt{x^2}=\sqrt1\)
\(\color{red}x = \pm 1\)
Method 2:
\(\color{aqua}x^2=1\)
\(\color{aqua} x^2-1=0\)
\(\color{aqua}x = {-0 \pm \sqrt{0^2-4(1)(-1)} \over 2(1)}= \pm 1\)
Either ways show that \(\sqrt 1\) have 2 answers, 1 and -1. But to state clearly which answer do we mean, we use \(\sqrt1\) to denote 1 and \(-\sqrt1\) to denote -1.
