June 28, 2022

# How to find Square root of a complex number

How to find Square root of a complex number

A number of the form

where and is called a complex number where is called as real part and is called imaginary part of complex number. To find the square root of a complex number, we will assume that . Then square  both the sides  and compare real and imaginary part to  find the value of and , which will give us the square root.

Formula for finding square root of a complex number:

The square root of  is

for      and

for

Proof. Let square root of . That is    where and in . Now square both the sides we get

Equating real and imaginary parts we get

Now

Solving (i) and (iii)  we get

.

Similarly . Since 2ab=y , it is clear that both and have the same sign when is positive and and   have different sign when is negative.  Therefore

for      and

for

Example : Find the square root of (-5-12i) .

Sol.  Here

Applying the formula for square root we get

( is negative)

Trick to find the square root of a complex number:

To find   , follow the following steps:

1. First find the number .
2.  Now factorise the given number in such a way that difference of square of these factors is equal to the real number .

Ex.  Find the square root of 7+4i.

Sol. Find the number   which is Equal to 12. Now factor 12 in such a way that difference of square of these factors is equal to the real number  .

12= 4×3   and . Therefore .

Ex.   Find the square root of .

Sol. Here    and such that

Thus

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