Let, √(5+12i)=x+iy On squaring both sides we get, 5+12i=(x+iy)2=(x2−y2)+i(2xy) On comparing real and imaginary parts on both sides we get (x2−y2)=5 and 2xy=12⇒xy=6 (x2+y2)=√(x2−y2)+4x2y2=√(5)2+4×36=√169=13(∵(x+y)2=(x−y)2+4xy) Now, (x2−y2)=5 and (x2+y2)=13 ⇒2x2=18 ⇒x2=9 ⇒x=±3 And, 9−y2=5 ⇒y2=4 ⇒y=±2 Hence, √(5+12i)=±(3+2i) Hence, option (2) is correct.