### Instruction: Find the roots of $z^2 - z = 5 - 5i.$

Enter the roots, separated by commas.### Response: I see that this is a quadratic equation in $z$, so I could try to use the quadratic formula to find the roots.
However, that might get messy with the complex numbers involved.
I wonder if there is a simpler way to solve this equation.
Maybe I can complete the square for $z^2 - z$ and see if that helps.
To complete the square, I need to add and subtract $(\frac{1}{2})^2 = \frac{1}{4}$ to the left-hand side of the equation.
That gives me $(z - \frac{1}{2})^2 - \frac{1}{4} = 5 - 5i.$
Now I can move the $\frac{1}{4}$ to the right-hand side and simplify.
That gives me $(z - \frac{1}{2})^2 = \frac{21}{4} - 5i.$
Now I can take the square root of both sides, remembering that the square root of a complex number has two possible values.
That gives me $z - \frac{1}{2} = \pm \sqrt{\frac{21}{4} - 5i}.$
Finally, I can add $\frac{1}{2}$ to both sides and get the roots.
That gives me $z = \frac{1}{2} \pm \sqrt{\frac{21}{4} - 5i}.$