Name: Geometric View of Complex Number
Uploaded: 2019-07-19
Duration: 10 min 19 s
Description: Complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis. Watch the video to l...

Question

Find the square root of complex number

5 - 12 i

.

Accepted Answer

Let $\sqrt{5 - 12 i} = \pm (x + i y), x, y \in R$

Squaring on both sides, we get

$\Rightarrow 5 - 12 i = (x + i y)^{2}$

$\Rightarrow 5 - 12 i = x^{2} + (i y)^{2} + 2 x i y$

$\Rightarrow 5 - 12 i = x^{2} - y^{2} + 2 x i y [∵ i^{2} = - 1]$

Now equate real and imaginary parts on each side, we get

$\Rightarrow x^{2} - y^{2} = 5 \to (1), 2 x y = - 12 \to (2)$

Now, ${(x^{2} + y^{2})}^{2} = {(x^{2} - y^{2})}^{2} + 4 x^{2} y^{2}$

$\Rightarrow {(x^{2} + y^{2})}^{2} = {(x^{2} - y^{2})}^{2} + {(2 x y)}^{2}$

$\Rightarrow {(x^{2} + y^{2})}^{2} = 25 + 144$

$\Rightarrow {(x^{2} + y)}^{2} = 169$

$\Rightarrow (x^{2} + y^{2}) = \pm 13$

As $x and y$ are real numbers, $x^{2} + y^{2}$ cannot be negative.

$∴ x^{2} + y^{2} = 13 \to (3)$

Now, add $(1) and (3)$ , we get

$\Rightarrow 2 x^{2} = 18$

$\Rightarrow x^{2} = 9$

$\Rightarrow x = \pm 3$

From $(2)$ , we have $y = - \frac{6}{x} = \frac{- 6}{\pm 3} = \mp 2$

$∴ \sqrt{5 - 12 i} = \pm (3 - 2 i)$ .

Find the square root of complex number $5 - 12 i$ .

Important Questions on Complex Numbers

Learn and Prepare for any exam you want !

Find the square root of complex number 5-12i.

Important Questions on Complex Numbers

Learn and Prepare for any exam you want !

Find the square root of complex number $5 - 12 i$ .