Complex solutions of the equation $x^{\frac{1}{2}}+1=0$ How can I find complex solutions to the equation $$x^{\frac{1}{2}}+1=0$$ 
Squaring gives x=1 but it's not the solution 


Squaring gives $\,x=1\,$ but it's not the solution

So you proved that $\,x^{1/2} = -1 \implies\, \big(x^{1/2}\big)^2 = (-1)^2 \implies x=1\,$, but $\,x=1\,$ does not verify the original equation (assuming that $\,x^{1/2}\,$ means the principal value of the complex square root). Therefore the original equation has no solutions, and there is nothing to further prove or look for.
