# Complex variables: solve for roots of $(z+\frac{i}{2})^3-i=0$

$(z+\frac{i}{2})^3-i=0$

Just got this problem on my Complex Variables hw and have no idea how to go about it. I know how to solve things like $z^n=w$, roots of unity etc but the $\frac{i}{2}$ is really throwing me off. Thanks!

• Hint: let $w = z+\frac i2$. – Théophile Jan 24 '17 at 22:00
• It's sufficient that after finding solutions $z^3=i$ add $-\dfrac{i}{2}$ to them. – Nosrati Jan 24 '17 at 22:03
• Calculate all three cubic roots of i. Then equate each of them to $z+\frac{i}{2}$. – Rafa Budría Jan 24 '17 at 22:04

$$z+\frac{i}{2}=w$$
and notice that $$w^3=i, \bar{w}^3=-i$$