Relations And Functions

• Question

# Find the square root of the following complex numbers :-5 + 12i

Solution

Let,  = x + iy                                                              ....(i)
Squaring both sides, we get
-5 + 12i = (x + iy)2
- 5 + 12i = x2 + i2y2 + 2xyi
- 5 + 12i = x2 - y2 + 2xyi                                                               ....(ii)
Equating real and uimaginary parts, we get
x2 - y2  = -5                                                                        ....(iii)
2xy = 12                                                                            ....(iv)
4x2y2 = 144
Also,                    (x2+ y2)2 = ((x2 - y2)2 + 4x2 y2
(x2+ y2)2 = 25 + 144 = 169
x2- y2  = 13                                                            ...(v)
(∵ x2- ycannot be negative for x, y  R)
From (iii) a nd (v), we have
x2- y2  = -5

x2+ y2 = 13
Adding, we get      2x2 = 8 x2 = 4 x = 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
or
Hence,