Question

Find the square root of $-5+12i$.

Answer 1

$-5+12i$

$\sqrt{a+ib}=\pm [\sqrt{\frac{a^2+b^2+a}{2}}+i\sqrt{\frac{a^2+b^2-a}{2}}]$

so $a=-5,b=12\Rightarrow a^2=25,b^2=144$

$\sqrt{-5+12i}=\pm [\sqrt{\frac{25+144+(-5)}{2}}+i\sqrt{\frac{25+144-(-5)}{2}}]$

$=\pm [\sqrt{\frac{\sqrt{169}-5}{2}}+i\sqrt{\frac{\sqrt{169}+5}{2}}]$

$=\pm [\sqrt{\frac{13-5}{2}}+i\sqrt{\frac{13+5}{2}}]$

$=\pm [\sqrt{\frac{8}{2}}+i\sqrt{\frac{18}{2}}]$

$=\pm [\sqrt{4}+i\sqrt{9}]$

$\sqrt{-5+12i}=\pm [2+3i].$