Question

Find root for complex number $5+12i$ ,$-5+12$

Answer 1

Methods for square root of a complex number
Let$z=\sqrt{5+12i}=x+iy$ squaring
$x^2-y^2+2ixy=5+12i$

$x^2-y^2=5,xy=6.$ solve By inspection$:5+12i$
Take half of coefficient of 'i' , i. e. $\frac{1}{2}$ (12)=6. Now decompose into two factors , squares of whose different is 5 i.e. 3 and 2 ... $3^2-2^2=5$
Hence$\sqrt{5+12i}=\pm (3+2i)$
$\therefore \sqrt{5-12i}=\pm (2+3i).$conjugate
and $\sqrt{-5+12i}=\pm (2+3i)$
and $\sqrt{-5-12i}=\pm (2-3i)$ conjugate
Here R.P.$=-5=2^2-3^2$ . Hence we chose $2\pm 3i,3\pm 2i$