Question

Stie cineva cum se rezolva?

Answer 1

Răspuns:

Explicație pas cu pas:

z²=3+4i,   evident zeste complex, deci z=x+yi, atunci

(x+yi)²=3+4i, ⇒x²+2xyi+(yi)²=3+4i, ⇒x²+2xy+y²·i²=3+4i, ⇒x²+2xyi+y²·(-1)=3+4i, ⇒x²-y²+2xyi=3+4i.

doua numere complexe sunt egale, daca au egale partile reale si imaginare, deci

$\left \{ {{x^{2}-y^{2}=3} \atop {2xy=4}} \right.~\left \{ {{x^{2}-y^{2}=3} \atop {xy=2}} \right.~\left \{ {{y=\frac{2}{x}} \atop {x^{2}-y^{2}=3}} \right.  ,~x^{2}-(\frac{2}{x} )^{2}=3~|*x^{2},~(x^{2})^{2}-3x^{2}-4=0,~Fie~x^{2}=t>0,~t^{2}-3t-4=0,~delta=9+16=25>0,~t_{1}=(3-5)/2=-1<0~si~t_{2}=(3+5)/2=4>0,~deci~x^{2}=4,~x_{1}=-\sqrt{4}=-2;~ x_{2}=+\sqrt{4}=+2;\\pentru~x=-2,~y=2/(-2)=-1\\pentru~x=2,~y=2/2=1\\Deci~z_{1}=-2-i;~z_{2}=2+i.$