# How to Find Square Root of Complex Number

Here you will learn what is square root and how to find square root of complex number with examples.

Let’s begin –

## How to Find Square Root of Complex Number

Let a + ib be a complex number such that $$\sqrt{a + ib}$$ = x + iy, where x and y are real numbers.

Then,

$$\sqrt{a + ib}$$ = x + iy

$$\implies$$ (a + ib) = $$(x + iy)^2$$

$$\implies$$ a + ib = $$(x^2 – y^2)$$ + 2ixy

On equating real and imaginary parts, we get

$$x^2 – y^2$$ = a                  ………….(i)

and,  2xy = b                        …………..(ii)

Now, $$(x^2 + y^2)^2$$ = $$(x^2 – y^2)$$ + $$4x^2y^2$$

$$\implies$$ $$(x^2 + y^2)^2$$ = $$a^2 + b^2$$

$$\implies$$ $$(x^2 + y^2)$$ = $$\sqrt{a^2 + b^2}$$               ………..(iii)

Solving the equations (i) and (iii), we get

$$x^2$$ = $$(1\over 2)$${$$\sqrt{a^2 + b^2} + a$$}   and  $$y^2$$ = $$(1\over 2)$${$$\sqrt{a^2 + b^2} – a$$}

$$\implies$$  x = $$\pm$$ $$\sqrt{{(1\over 2)}{\sqrt{a^2 + b^2} + a}}$$  and  y = $$\pm$$ $$i\sqrt{{(1\over 2)}{\sqrt{a^2 + b^2} – a}}$$

If b is positive, then by equation (ii), x and y are of the same sign.

Hence, $$\sqrt{a + ib}$$ = $$\pm$$ [$$\sqrt{{(1\over 2)}{\sqrt{a^2 + b^2} + a}}$$ + $$i\sqrt{{(1\over 2)}{\sqrt{a^2 + b^2} – a}}$$]

If b is negative, then by equation (ii), x and y are of the different signs.

Hence, $$\sqrt{a + ib}$$ = $$\pm$$ [$$\sqrt{{(1\over 2)}{\sqrt{a^2 + b^2} + a}}$$ – $$i\sqrt{{(1\over 2)}{\sqrt{a^2 + b^2} – a}}$$]

Example : Find the square root of 7 – 24i.

Solution : Let $$\sqrt{7 – 24i}$$ = x + iy. Then,

(7 – 24i) = $$(x + iy)^2$$

$$\implies$$ 7 – 24i = $$(x^2 – y^2)$$ + 2ixy

On equating real and imaginary parts, we get

$$x^2 – y^2$$ = 7                  ……….(i)

and, 2xy = -24                     …………..(ii)

Now,  $$(x^2 + y^2)^2$$ = $$(x^2 – y^2)^2$$ + $$4x^2y^2$$

$$\implies$$ $$(x^2 + y^2)^2$$ = 49 + 576 = 625

$$\implies$$ $$x^2 + y^2$$ = 25            ………..(iii)

On Solving equation (i) and (iii), we get

$$x^2$$ = 16   and  $$y^2$$ = 9

$$\implies$$ x = $$\pm 4$$   and  y = $$\pm 3$$

From (ii), 2xy is negative. So, x and y are of opposite signs.

$$\therefore$$   (x = 4 and y = -3)  or,  (x = -4 and y = 3)

Hence,  $$\sqrt{7 – 24i}$$ = $$\pm$$ (4 – 3i)