Zadanie 1 [latex]z^2-(5-i)cdot z+8+4i=0\ Delta=(5-i)^2-4(1)(8+4i)=25-10i+i^2-32-16i=\=-8-26i\ sqrt{Delta}=sqrt{-8-26i}\\ oxed{z_1= frac{5-i-sqrt{-8-26i}}{2}\ z_2= frac{5-i+sqrt{-8-26i}}{2} }[/latex] Zadanie 2 [latex]z=(-1+i)^{19}\ \x=-1+i\ x=|x|(cosphi+isinphi)= sqrt{2}( cosphi+isinphi ) \cosphi = - frac{sqrt2}{2} ^o sinphi = frac{sqrt2}{2}\ phi=135^o \\ x^{19}=z=|x|^{19}(cos(19phi)+isin(19phi))= (sqrt{2})^{19}(cos(45^o)+isin(45^o))\ z= (sqrt{2})^{19}(frac{ sqrt{2} }{2} +i frac{ sqrt{2} }{2} )=frac{ sqrt{2}^{20} }{2} +i frac{ sqrt{2}^{20} }{2} =oxed{512+512i}\ [/latex] Część rzeczywista to [latex]512[/latex] Część urojona to [latex]512[/latex] Zadanie 3 [latex]z= frac{4-3i}{1+2i}=frac{4-3i}{1+2i}cdot frac{1-2i}{1-2i}=frac{(4-3i)(1-2i)}{5}= frac{4-8i-3i+6i^2}{5} =\=frac{-2-11i}{5}=oxed{- frac{2}{5}- frac{11}{5}i } [/latex] Część rzeczywista to [latex] -frac{2}{5} [/latex] Część urojona to [latex]- frac{11}{5} [/latex]
