Question

Find the square root of -12+6i

Answer 1

$\textbf{Given:}$

$\text{Complex number is -12+6\,i }$

$\textbf{To find:}$

$\text{Square root of -12+6\,i }$

$\textbf{Solution:}$

$\textbf{We know that,}$

$\textbf{The square root of \bf\,z=a+i\,b is}$

$\bf\sqrt{a+i\,b}=\pm[\sqrt{\dfrac{|z|+a}{2}}+i\dfrac{b}{|b|}\sqrt{\dfrac{|z|-a}{2}}]$

$\text{Let}\;z=-12+6\,i$

$\text{Here,}\,a=-12\;\text{and}\;b=6$

$|z|=\sqrt{a^2+b^2}$

$|z|=\sqrt{144+36}$

$|z|=6\sqrt{5}$

$\text{Now,}$

$\sqrt{-12+6\,i}=\pm[\sqrt{\dfrac{|z|+a}{2}}+i\dfrac{b}{|b|}\sqrt{\dfrac{|z|-a}{2}}]$

$\sqrt{-12+6\,i}=\pm[\sqrt{\dfrac{6\sqrt{5}-12}{2}}+i\dfrac{6}{|6|}\sqrt{\dfrac{6\sqrt{5}+12}{2}}]$

$\sqrt{-12+6\,i}=\pm[\sqrt{3\sqrt{5}-6}+i\,\sqrt{3\sqrt{5}+6}]$

$\textbf{Answer:}$

$\textbf{Square root of -12+6i are}$

$\bf\pm[\sqrt{3\sqrt{5}-6}+i\,\sqrt{3\sqrt{5}+6}]$