Question

The square root of $3+4i$ is

• A. $\pm (2-i)$
• B. $\pm (2+i)$
• C. $\pm (3+i)$
• D. $\pm (3-i)$

Answer 1

The correct option is B $\pm (2+i)$

Let x+iy be sq. root of 3+4i then

$(x+iy{)}^2=3+4i$

$\Rightarrow x^2-y^2+2xyi=3+4i$

on comparing we get

$x^2-y^2=3$ _________ (1) and 2xy=4.

we know $(x^2+y^2{)}^2=(x^2-y^2{)}^2+(2xy{)}^2$

$(x^2+y^2{)}^2=9+16=25$

$x^2+y^2=5$ ________ (2)

From equation (1) & (2) we get $2x^2=8$

$x^2=4[x=\pm 2]$

and 2xy=4 $y=\pm 1$

Therefore square root of 3+4i is $\pm (2+i)$