Paradox?

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.