The correct way to solve this equation :$\sqrt{-|x|}+\sqrt{|x|}=0$ in $\mathbb{R}$?

Question

it is clear at all that $x=0$ is the only solution of this equation :$\sqrt{-|x|}+\sqrt{|x|}=0$ But am not confident in my way of solving : I have used the definition of absolute value but i have $\sqrt{-|x|}$ is not defined at x negative and positive it is defined only at $x=0$ which it is the solution, I have used other method such that i took $\sqrt{-|x|}+\sqrt{|x|}=0$ implies that $\sqrt{-|x|}=-\sqrt{|x|}=0$ this gives by squarting bot side of the equality :$-|x|=|x|$ implies $x=0$ , But this method seems to me is not logic because I didn't defined $\sqrt{-|x|} $, Really this equation dosn't have a way , the solution still clear , But I want to follow any analytical logic way to get that solution ?

Answer 1

The term $\sqrt{-|x|}$ is not defined over $\mathbb{R}$ unless $x=0$. Since a square root of a negative number does not make sense over reals. Hence, the solution, if it does exist, must be $0$. Thus you plug in $0$, it works, and you are done!

Answer 2

Both roots should have non-negative values under them. Thus, the only value that makes both roots to contain a non-negative value simultaneously is 0. And it's easy to check if 0 is the solution for the equation.