Find the square roots of -15 + 8i.

Step 1. Three equations. For $(x + iy)^2 = -15 + 8i$: $x^2 - y^2 = -15$, $2xy = 8$ and $x^2 + y^2 = \sqrt{225 + 64} = 17$.

Step 2. Add and subtract. $2x^2 = 2$ and $2y^2 = 32$, so $x = \pm 1$ and $y = \pm 4$.

Step 3. The signs. $2xy = 8$ is positive, so $x$ and $y$ have the same sign.

Step 4. Check. $(1 + 4i)^2 = 1 + 8i - 16 = -15 + 8i$.