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Square Root of complex Number [Short Trick] – 7tricks

Today, we will tell you the super-duper trick to find the square root of any complex number in just 20 seconds only.

Trick to Find the Square Root of any complex Number

First, we talk about the method of finding the square root of a complex number.

Let a complex number (a+ib) and we have to find the square root of (a+ ib)
Assume (a+ib) = (x+iy)

Squaring both sides we get
a + ib = x2 + i2y2+ 2ixy
Now, equate the real parts of both side also, imaginary parts of both side

a = x2 – y2  &  b = 2xy

By solving this
We can find the value of x and y
Then the complex number we will get is x+iy so it is the square root of (a + ib).

It’s a long method and takes so much time to get the answer, As we talking about the trick we have to solve these types of questions in just a few seconds.

TRICK

To calculate the square root of (a+ib)
Let it is equal to (x+iy) and we will get values of x & y by these shortcut formulas
x2 = square root ( a2+ b2) + a/2
y
y2 = square root (a2 + b2) – a/2
BY this shortcut formula we will get the value of x and y instantly.

Let us try out with some examples:- 
Example 1
Que: Find the square root of (3+4i).
Sol: Let √(3+4i) is equal to (x+iy)
x2 = [ √(32+42) + 3] ÷ 2
x2 =  (5+3) ÷ 2
x2 = 4 or x = 2
So, we got the value of x is equal to 2.

y2[(32 + 42) – 3] ÷ 2
y2 = (5-3) ÷ 2
y2 = 1 or y = 1
So, we got the value of y is equal to 1.
The square root of complex number a + ib = x + iy
Square root (3+ 4i)  = ±(2 + i)

Note:- One point is to remember here is that if we have +ve sign between the real part and imaginary part [e.g. (a+ib)]. Then, for (x+iy) here sign is also positive.
Example 2
Que: Find the square root of (3 – 4i).
SolLet square root (3 – 4i)  equal to  (x + iy)
x2 = [ √{32+(-4)2} + 3] ÷ 2
x= 4 or x = 2
So the value of x is equal to 2
y2[ √{32+(-4)2} – 3] ÷ 2
y2 = 1 or y = 1
So the value of y is equal to 1
Thus,
Square root (3 -4i) = ±(2 – i)

Example 3
Que: Find the square root of (-3 – 4i).
Sol
Let square root (-3 – 4i)  equal to  x + iy
x2 = [ √{(-3)2+(-4)2} +(-3)] ÷ 2
x= 1 or x = 1

So the value of x is equal to 1

y2 = [ √{(-3)2+(-4)2} -(-3)] ÷ 2
y2 = 4 or y = 2

So the value of y is equal to 2 .

Thus,
Square root of (3 -4i) = ±(1 – 2i)

This method is very easy and very fast, if your calculation is fast then it will take some seconds to get the answer. You can solve any type of questions related to finding the square root of a complex number by this shortcut.

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