Find the square root of the following complex number: `-7-24 i`

Explanation

Let `sqrt(-7-24i)=a+ ib`

`-7-24i=(a+ib)^2=a^2-b^2+2iab`

So, `a^2-b^2=-7` and `2ab=-24=>ab=-12=>b=frac{-12}{a}`

`a^2-frac{144}{a^2}=-7`

`a^4+7a^2-144=0=>(a^2-9)(a^2+16)=0`

`a=+-3`

`a=frac{-12}{a}=+-4`

So, `a=3, b=-4` and `a=-3, b=4`
...