Find the square root of a complex number

 

Question

 

               Find the square root of   8 – 6i

 

 

 

First method

 

                  Let        z2 = (x + yi)2  = 8 – 6i

          \    (x2 – y2) + 2xyi     = 8 – 6i

          Compare real parts and imaginary parts,

                                          x2 – y2 = 8                                      (1)

                                          2xy = -6                                          (2)

          Now, consider the modulus:         |z|2 = |z2|

                          \    x2 + y2 = Ö(82 + 62) = 10                      (3)

          Solving (1) and (3), we get           x2 = 9   and    y2 = 1

                                                                  x = ±3  and          y = ±1

          From (2), x and y are of opposite sign,

                  \ (x = 3 and y = -1)    or   (x = -3 or y = 1)

                  \ z =  ± (3 – i)

 

 

Second method

 

 

                  Please note that :

                        

 

                  Therefore,