Relations and Functions

Relations and Functions

Question

Find the square root of the following complex numbers :

-5 + 12i

Answer

Let,  space space space space space space space space square root of negative 5 plus 12 straight i end root = x + iy                                                              ....(i)
Squaring both sides, we get
-5 + 12i = (x + iy)2
rightwards double arrow   - 5 + 12i = x2 + i2y2 + 2xyi
rightwards double arrow - 5 + 12i = x2 - y2 + 2xyi                                                               ....(ii)
Equating real and uimaginary parts, we get
            x2 - y2  = -5                                                                        ....(iii)
              2xy = 12                                                                            ....(iv)
rightwards double arrow      4x2y2 = 144 
Also,                    (x2+ y2)2 = ((x2 - y2)2 + 4x2 y2
rightwards double arrow                      (x2+ y2)2 = 25 + 144 = 169
rightwards double arrow                      x2- y2  = 13                                                            ...(v)            
                                                  (∵ x2- ycannot be negative for x, y  space space element ofR)
From (iii) a nd (v), we have
                    x2- y2  = -5
                
                     x2+ y2 = 13
Adding, we get      2x2 = 8 rightwards double arrowx2 = 4 rightwards double arrow x = plus-or-minus 2
When x = 2, from (iii), 2(2) (y) = 12 or y = 3
When x = 2, from (iii), 2(-2)y = 12 or y = -3
Putting in (i)  we have space space space space space space space space square root of negative 5 space plus space 12 straight i end root space space equals space 2 space plus space 3 straight i
or        space space space space space space space space space space space space square root of negative 5 plus 12 straight i end root space equals space minus 2 minus 3 straight i space equals space minus left parenthesis 2 plus 3 straight i right parenthesis
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