FIND THE SQUARE ROOT OF COMPLEX NUMBER

Find the square root of complex number :

Here we are going to see how to find the square root of complex number.

Let a + ib be a complex number such that √a + ib  =  x + iy.

Let us look in to some example problems to understand the concept.

Example 1 :

Find the square root of the following 

7 - 24i

Solution :

Let √7 - 24i  =  x + iy, then

√(7 - 24i)  =  x + iy

Taking squares on both sides

7 - 24i  =  (x + iy)2

7 - 24i  =  x2 + (iy)2 + 2(x)(iy)

7 - 24i  =  x2 - y2 + i 2xy

x2 - y =  7   ------(1)     2xy  =  -24  ------(2)

We have a formula,

(x2 + y2)=  (x2 - y2)2 + 4x2y2

=  (x2 - y2)2 + (2xy)2

=  72 + (-24)2

 (x2 + y2)2  =  49 + 576  ==>  625

 (x2 + y2)  =  √625  ==>  25

x2 + y2  =  25    ------(3)

(1)  +  (3) ==>   x2 - y2 x2 + y2  =  25 + 7

2x2  =  32

x2  =  16 ==>  x  =  ± 4

x  =  4

2(4)y  =  -24

y =  -24/8  ==> -3

x  =  -4

2(-4)y  =  -24

y =  -24/(-8)  ==> 3

Hence the square root of the given complex number are

4 - 3i or (-4 + 3i)

Example 2 :

Find the square root of the following 

-15 - 8i

Solution :

Let √-15 - 8i  =  x + iy, then

√(-15 - 8i)  =  x + iy

Taking squares on both sides

-15 - 8i  =  (x + iy)2

-15 - 8i  =  x2 + (iy)2 + 2(x)(iy)

-15 - 8i  =  x2 - y2 + i 2xy

x2 - y =  -15  ------(1)     2xy  =  -8  ------(2)

We have a formula,

(x2 + y2)2  =  (x2 - y2)2 + 4x2y2

=  (x2 - y2)2 + (2xy)2

=  (-15)+ (-8)2

 (x2 + y2)2  =  225 + 64  ==>  289

 (x2 + y2)  =  √289  ==>  17

x2 + y2  =  17    ------(3)

(1)  +  (3) ==>   x2 - yx2 + y2  =  -15 + 17 

2x2  =  2

x2  =  1 ==>  x  =  ± 1

x  =  1

2(1)y  =  -8

y =  -8/2  ==> -4

x  =  -1

2(-1)y  =  -8

y =  -8/(-2)  ==> 4

Hence the square root of the given complex number are

1 - 4i or (-1 + 4i)

Example 3 :

Find the square root of the following 

-8 - 6i

Solution :

Let √-8 - 6i  =  x + iy, then

√(-8 - 6i)  =  x + iy

Taking squares on both sides

-8 - 6i  =  (x + iy)2

-8 - 6i  =  x2 + (iy)2 + 2(x)(iy)

-8 - 6i  =  x2 - y2 + i 2xy

x2 - y =  -8  ------(1)     2xy  =  -6  ------(2)

We have a formula,

(x2 + y2)2  =  (x2 - y2)2 + 4x2y2

=  (x2 - y2)2 + (2xy)2

=  (-8)+ (-6)2

 (x2 + y2)2  =  64 + 36  ==>  100

 (x2 + y2)  =  √100  ==>  10

x2 + y2  =  10    ------(3)

(1)  +  (3) ==>   x2 - yx2 + y2  =  -8 + 10 

2x2  =  2

x2  =  1 ==>  x  =  ± 1

x  =  1

2(1)y  =  -6

y =  -6/2  ==> -3

x  =  -1

2(-1)y  =  -6

y =  -6/(-2)  ==> 3

Hence the square root of the given complex number are

1 - 3i or (-1 + 3i)

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