Răspuns:
61)
Notăm [tex]z^2=t[/tex]
[tex]t^2-2(2+i)t+8i=0[/tex]
[tex]\Delta=4(4+4i+i^2)-32i=4(4-4i+i^2)=4(2-i)^2[/tex]
[tex]t_1=\displaystyle\frac{2(2+i)-2(2-i)}{2}=2i\\t_2=4[/tex]
[tex]z^2=2i=1+2i+i^2=(1+i)^2\Rightarrow z=\pm(1+i)\\z^2=4\Rightarrow z=\pm 2[/tex]
62)
Din enunț avem
[tex]z\cdot\bar{z}=u\cdot\bar{u}=1\Rightarrow\bar{z}=\displaystyle\frac{1}{z}, \ \bar{u}=\frac{1}{u}[/tex]
Atunci
[tex]\displaystyle\overline{\left(\frac{z+u}{1+zu}\right)}=\frac{\bar{z}+\bar{u}}{1+\bar{z}\bar{u}}=\frac{\displaystyle\frac{1}{z}+\frac{1}{u}}{1+\displaystyle\frac{1}{z}\frac{1}{u}}=\frac{u+z}{1+uz}[/tex]
Deci expresia este număr real.
Explicație pas cu pas:
