Find the square root of the following complex number: `-5+12 i`

Options

A. `2-3i`
B. `2+3i`
C. `-2+3i`
D. `sqrt(-5)+sqrt(12i)`

Answer

B

Explanation

Let `sqrt(-5+12i)=x+iy`
`implies(x+iy)^(2)=-5+12i`
`impliesx^(2)-y^(2)+i2xy=-5+12i`
`x^(2)-y^(2)=-5and 2xy=12impliesxy=6`
`(x^(2)-y^(2))^(2)+4x^(2)y^(2)=(x^(2)+y^(2))^(2)`
`implies(-5)^(2)+4xx(6)^(2)=(x^(2)+y^(2))^(2)`
`implies(x^(2)+y^(2))^(2)=25+144=169`
`impliesx^(2)+y^(2)=+-13 and x^(2)-y^(2)=-5`
Adding both `2x^(2)=+-13-5=8or -18`
`impliesx^(2)=4(-"ve discarded)"`
`impliesx=+-2and y^(2)=+-13-4=9or -13"Discard"y^(2)=-13,yy^(2)=9`
`impliesy=+-3`
`impliesx+iy=+-(2+3i)`
`impliessqrt(-5+12i)=(2+3i)`