Solve the quadratic equation z2 + 12z + 52 = 0 for z.

$$z^2 + 12z + 52 = 0$$ $$z^2 + 12z = -52$$

To complete the square, take half of the coefficient of $z$ (which is 12), square it, and add it to both sides of the equation.
Half of 12 is 6.
$6^2 = 36$.

$$z^2 + 12z + 36 = -52 + 36$$

$$\sqrt{(z + 6)^2} = \sqrt{-16}$$ $$z + 6 = \pm \sqrt{16} \cdot \sqrt{-1}$$ $$z + 6 = \pm 4i$$

$$z = -6 \pm 4i$$

The solutions are $z = -6 + 4i$ and $z = -6 - 4i$.