Q.Find the square roots of $-15+8i$.

Set $\sqrt{-15+8i}=x+yi$; use $x^{2}-y^{2}=-15$, $2xy=8$ and $x^{2}+y^{2}=\sqrt{(-15)^{2}+8^{2}}=17$.

Let $\sqrt{-15+8i}=x+yi$. Squaring gives

so

From moduli, $x^{2}+y^{2}=\sqrt{(-15)^{2}+8^{2}}=\sqrt{225+64}=\sqrt{289}=17.$

Combine with $x^{2}-y^{2}=-15$:

Since $2xy=8>0$, $x$ and $y$ share the same sign: …

Mixing up which of $x^{2},y^{2}$ gets the $+a$ vs $-a$: because $a=-15$ is negative, $x^{2}=\tfrac12(17-15)=1$ is the small one and $y^{2}=\tfrac12(17+15)=16$ the …