The square roots of 5-2 14 i are - Vidyadip

MCQ

The square roots of $5-2 \sqrt{14}$ i are

A
$\sqrt{7}+\sqrt{2} i ,-\sqrt{7}+\sqrt{2} i$
✓
$\sqrt{7}-\sqrt{2} i ,-\sqrt{7}+\sqrt{2} i$
C
$\sqrt{7}-\sqrt{2} i,-\sqrt{7}-\sqrt{2} i$
D
$-\sqrt{7}-\sqrt{2} i ,-\sqrt{7}+\sqrt{2} i$

✓

Answer

Correct option: B.

$\sqrt{7}-\sqrt{2} i ,-\sqrt{7}+\sqrt{2} i$

(B)
Here, b < 0
Square root of $z=a+i b$ is
$\begin{aligned}
\sqrt{a+ib} & = \pm\left[\sqrt{\frac{|z|+a}{2}}+i \sqrt{\frac{|z|-a}{2}}\right], \text { for } b>0 \\
& = \pm\left[\sqrt{\frac{|z|+a}{2}}-i \sqrt{\frac{|z|-a}{2}}\right], \text { for } b<0\end{aligned}$
$\therefore \sqrt{5-2 \sqrt{14} i }= \pm\left[\sqrt{\frac{9+5}{2}}- i \sqrt{\frac{9-5}{2}}\right]$
$= \pm(\sqrt{7}-\sqrt{2} i)$

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