Find square roots of $8 - 15i$

Question

Find the square roots of: $8-15i.$

Could I get some working out to solve it?

Also what are different methods of doing it?

Answer 1

You might start with solving $$ (x + i y)^2 = 8 - 15 i $$

component-wise (real part vs imaginary part), which leads to two equations in the two unknowns $x$ and $y$.

More general is using Euler's formula $e^{i \phi} = \cos \phi + i \sin \phi$, using polar coordinates $(r, \phi)$, $z = r e^{i \phi}$ with $z = x + i y$, $r = |z| = \sqrt{x^2 + y^2}$ and $\tan \phi = y / x$.