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Q. The square root of the number $(5+12i)$ is

Complex Numbers and Quadratic Equations

Solution:

If $z = 5 + 12i$, then $\left|z\right|=13$
So, square root of $z =\pm\left[\sqrt{\frac{\left|z\right|+a}{2}}+i\sqrt{\frac{\left|z\right|-a}{2}}\right]$,
where $a = 5 ($real part of $z)$
$=\pm\left[\sqrt{\frac{13+5}{2}}+i\sqrt{\frac{13-5}{2}}\right]$
$=\pm\left[3+2i\right]$