Question

Sa se determine numerele complexe z stiind ca z, z^2 , 1+z au același modul
dau coroana!! ​

Answer 1

$\displaystyle |z|=|z^2| \Leftrightarrow |z|=|z|^2 \Leftrightarrow |z| \in \{0,1\}. \\ \\ Daca~|z|=0,~rezulta~z=0,~care~nu~convine~pentru~ca~in~acest~caz \\ \\ am~avea~|1+z|=1 \\ \\ Daca~|z|=1,~atunci~\overline{z}=\frac{1}{z}. \\ \\ Din~ipoteza~rezulta~|1+z|=1,~deci~\overline{1+z} \cdot (1+z)=1 \Leftrightarrow \\ \\ \Leftrightarrow \left(1+ \frac{1}{z} \right)(1+z)=1 \Leftrightarrow z+\frac{1}{z}+1=0 \Leftrightarrow z^2+z+1=0.$

$\displaystyle Rezolvand~ecuatia~obtinem~z \in \left\{ - \frac{1}{2}+ \frac{\sqrt{3}}{2}i, -\frac{1}{2}- \frac{\sqrt{3}}{2}i\right\}. \\ \\ Se~verifica~fiecare~solutie~si~se~constata~ca~amandoua~convin. \\ \\ Observatie:~Solutiile~gasite~sunt~radacinile~nereale~de~ordin~3~ale~ \\ \\ unitatii.$