Question

Soluțiile ecuației z^2+|z|=0, z∈ℂ sunt:

Answer 1

$z^2 \to (a+bi)^2 = a^2 +2abi-b^2 \\\\ |z| \to \sqrt{a^2+b^2} \\\\\\ a^2 +2abi-b^2+\sqrt{a^2+b^2}=0 \\\\ \sqrt{a^2+b^2}= b^2-2abi-a^2 \\\\ \left \{ {{-2ab=0 }} \\ \\ \\ \atop {\sqrt{a^2+b^2}=b^2-a^2}} \right. \\\\\ -2ab=0 \ \ ==\ \textgreater \ \ \ I)a=0 \ \ \ \vee \ \ II)b=0 \\\\\\\ I)a=0 \\\\ \sqrt{b^2}=b^2\ \ |^2 \\\\ b^2=b^4 \\\\ b^4-b^2=0 \\\\ b^2(b^2-1)=0 \ \ \ ===\ \textgreater \ b=0 \ \ \ \vee \ \ \ b^2-1=0 \\\\ b^2-1=0 \ \ \ ==\ \textgreater \ b^2=1 \longrightarrow b= \pm 1 \\\\\\ z_1= 0 \ \ \ ; \ \ \ z_2=i \ \ \ ; \ \ \ z_3= -i$

$II) b=0 \\\\ \sqrt{a^2}= -a^2 \ \ \ 'F' \\\\\\ \boxed{S_f \in \underline{\{ z_1=0 \ \ \ ; \ \ \ z_2=i \ \ \ ; \ \ \ z_3= -i \}}}$