Calculate an estimate of a square root of 119 + 120i

Answer:Step-by-step explanation:Givenz=119+120 iLet [tex]\sqrt{119+120 i}=p+iq[/tex]Squaring both sides[tex]119+120 i=p^2-q^2+2ipq[/tex]Comparing real and imaginary partRe(LHS)=Re(RHS)[tex]119=p^2-q^2[/tex]-----------1comparing Im(LHS)=Im(RHS)120=2pq[tex]q=\frac{60}{p}[/tex]Substitute q in 1[tex]119=p^2-(\frac{60}{p})^2[/tex][tex]p^4-119p^2-(68)^2=0[/tex]Let [tex]x=p^2[/tex][tex]x^2-119x-4624=0[/tex][tex]x=frac{119\pm \sqrt{119^2+4\times 4624}}{2}[/tex][tex]x=\frac{119\pm 180.71}{2}[/tex]we take only Positive value because [tex]p^2=x[/tex]x=149.85 [tex]p^2=149.85[/tex]thus [tex]p=\pm 12.24[/tex][tex]q=\mp 4.90[/tex]thus [tex]\sqrt{119+120 i}=\pm (12.24+i 4.90)[/tex]