Basic math paradox

Question

1. $1 = 1$
2. $ 1^2 = 1$ and $(-1)^2 = 1$
3. Therefore, $1^2 = (-1)^2$
4. Square root both sides $\sqrt{1^2} = \sqrt{(-1)^2}$
5. Therefore, $-1 =1$

This is an obvious paradox, but I don't know how to approach solving it

Answer 1

This is not obvious and wrong. You applied that "if $f(x)=f(y)$, then $x=y$ for all $x,y$ in $\operatorname{dom} f$" between step 4 and 5. However, it is true if and only if $f$ is one-to-one. $f(x)=\sqrt{x^2}$ is not one-to-one.