Find the square root of complex number 3 + 4i

Strategy: write down an equation satisfied by the square root, and solve it algebraically.  Method:  square root x+iy  satisfies (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).