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if-z-2-4-z-2i-5-10i-find-Re-Z-Im-Z-




Question Number 146988 by mathdanisur last updated on 16/Jul/21
if   ((z^2  + 4)/(z - 2i)) = −5 + 10i  find   Re(Z) + Im(Z) = ?
$${if}\:\:\:\frac{\boldsymbol{{z}}^{\mathrm{2}} \:+\:\mathrm{4}}{\boldsymbol{{z}}\:-\:\mathrm{2}\boldsymbol{{i}}}\:=\:−\mathrm{5}\:+\:\mathrm{10}\boldsymbol{{i}} \\ $$$${find}\:\:\:{Re}\left({Z}\right)\:+\:{Im}\left({Z}\right)\:=\:? \\ $$
Commented by aliibrahim1 last updated on 17/Jul/21
z^2 +4=(z−2i)(z+2i)  (((z−2i)(z+2i))/(z−2i))=−5+10i  z+2i=−5+10i  z=−5+8i  Re(z)=−5  Im(z)=8  −5+8=3
$${z}^{\mathrm{2}} +\mathrm{4}=\left({z}−\mathrm{2}{i}\right)\left({z}+\mathrm{2}{i}\right) \\ $$$$\frac{\left({z}−\mathrm{2}{i}\right)\left({z}+\mathrm{2}{i}\right)}{{z}−\mathrm{2}{i}}=−\mathrm{5}+\mathrm{10}{i} \\ $$$${z}+\mathrm{2}{i}=−\mathrm{5}+\mathrm{10}{i} \\ $$$${z}=−\mathrm{5}+\mathrm{8}{i} \\ $$$${Re}\left({z}\right)=−\mathrm{5} \\ $$$${Im}\left({z}\right)=\mathrm{8} \\ $$$$−\mathrm{5}+\mathrm{8}=\mathrm{3} \\ $$
Commented by mathdanisur last updated on 17/Jul/21
thank you Ser
$${thank}\:{you}\:{Ser} \\ $$
Commented by aliibrahim1 last updated on 17/Jul/21
you are welcome
$${you}\:{are}\:{welcome}\: \\ $$

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