[tex]\displaystyle Rezolvam~ecuatia~z^2=-i~(*). \\ \\ Notam~z=a+bi,~a,b \in \mathbb{R}. \\ \\ (*)~devine~a^2-b^2+2abi=-i. \\ \\ Rezulta~a^2-b^2=0~si~2ab=-1. \\ \\ Acum~din~a^2=b^2,~avem~doua~cazuri:~a=b~sau~a=-b. \\ \\ Primul~nu~convine~(a=b \Rightarrow 2ab=2a^2\ \textgreater \ 0) \\ \\ a=-b \Rightarrow -2a^2=-1 \Rightarrow a^2= \frac{1}{2}. \\ \\ a= \frac{1}{ \sqrt{2} } \Rightarrow b=- \frac{1}{ \sqrt{2} }. \\ \\ a= -\frac{1}{ \sqrt{2}} \Rightarrow b= \frac{1}{ \sqrt{2}}.[/tex]
[tex]\displaystyle Deci~z \in \left \{ \frac{1}{\sqrt{2}}- \frac{1}{\sqrt{2}}i~,~ -\frac{1}{\sqrt{2}}+ \frac{1}{ \sqrt{2}}i\right \}.[/tex]
