4.6 Square Root of a Complex Number

Let $z = -7 - 24i$. $|z| = \sqrt{(-7)^2 + (-24)^2} = \sqrt{49 + 576} = \sqrt{625} = 25$. $a = -7, b = -24$. Since $b < 0$, $x$ and $y$ have opposite signs. $x^2 = \frac{25 + (-7)}{2} = \frac{18}{2} = 9 \implies x = 3$ $y^2 = \frac{25 - (-7)}{2} = \frac{32}{2} = 16 \implies y = 4$ Roots: $\pm (3 - 4i)$.