Question

Find the square root of $3+4i$ ?

Answer 1

(x+iy)² = 3 + 4i.

Expand: x²-y² +2xyi = 3+4i.

Comparing coefficients gives: x²-y² =3 and 2xy =4.

Then substitute y: x² -4/x² = 3.

Rearrange to get quadratic in x² : (x² +1)(x² -4) = 0.

x can't be imaginary (by definition) so x= +/- 2.

Plug in to equation 2xy = 4

get y = +/- 1. So square root is +/- (2+i).