Question

Find the square root of −5 − 12𝑖

Answer 1

$-5-12i=\left( a+ib \right) ^2\\-5-12i=a^2-b^2+2abi\\\left\{ \begin{array}{c} a^2-b^2=-5\\ 2ab=-12\\\end{array} \right. \Rightarrow \left\{ \begin{array}{c} b=-\frac{6}{a}\\ a^2-\frac{36}{a^2}=-5\\\end{array} \right. \Rightarrow \left\{ \begin{array}{c} b=-\frac{6}{a}\\ a^4+5a^2-36=0\\\end{array} \right. \Rightarrow \left\{ \begin{array}{c} b=-\frac{6}{a}\\ a=\pm 2\\\end{array} \right. \Rightarrow \\\Rightarrow \left\{ \begin{array}{c} \sqrt{-5-12i}_1=2-3i\\ \sqrt{-5-12i}_2=-2+3i\\\end{array} \right.$