Question

Find the square root of $8-6i$.

• A. $\pm (3-i)$
• B. $\pm (4+3i)$
• C. $\pm \sqrt{3}\left(1-3i\right)$
• D. None of these

Answer 1

Answer: (A)

$\pm (3-i)$

Let $z^2=(x+yi{)}^2=8-6i$

Therefore, $(x^2-y^2)+2xyi=8-6i$

Compare real parts and imaginary parts,

$x^2-y^2=8$ .....(1)

$2xy=-6$ .....(2)

Now, consider the modulus : $|z{|}^2=|z^2|$

Therefore, $x^2+y^2=\sqrt{8^2+6^2}=10$ ......(3)

Solving 1 and 3, we get,

$x^2=9$ and $y^2=1$

$x=\pm 3$ and $y=\pm 1$

From 2 $x$ and $y$ are of opposite sign.

Therefore, $(x=3,y=-1)$ or $(x=-3,y=1)$

Therefore, $z=\pm (3-i)$