# How do you solve z^2 +12 = -z?

May 22, 2018

-1/2±sqrt(47)/2i

#### Explanation:

Use the quadratic formula

${z}^{2} + 12 = - z \iff {z}^{2} + z + 12 = 0$

z=(-1±sqrt(1^2-4(12)(1)))/2=-1/2±sqrt(47)/2i

Alternatively, you could complete the square,

${z}^{2} + z + 12 = 0$
${z}^{2} + z + \frac{1}{4} + \frac{47}{4} = 0$
${\left(z + \frac{1}{2}\right)}^{2} = - \frac{47}{4}$
$z + \frac{1}{2} = \pm \frac{\sqrt{47}}{2} i$
$z = - \frac{1}{2} \pm \frac{\sqrt{47}}{2} i$

Which yields the same result (the quadratic formula is derived from completing the square)