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

Enter the roots, separated by commas.### Response: I notice that this is a quadratic equation in $z$, so I could try to use the quadratic formula to find the roots.
However, that might be messy, since the coefficients are complex numbers.
I wonder if there is a simpler way to solve this equation.
I notice that the left-hand side is a difference of squares, so I can rewrite it as $(z - \frac{1}{2})^2 - \frac{1}{4}.$
Then, I can move the constant term to the right-hand side and get $(z - \frac{1}{2})^2 = 5 - 5i + \frac{1}{4}.$
Now, I can take the square root of both sides and get $z - \frac{1}{2} = \pm \sqrt{5 - 5i + \frac{1}{4}}.$
Finally, I can add $\frac{1}{2}$ to both sides and get $z = \frac{1}{2} \pm \sqrt{5 - 5i + \frac{1}{4}}.$
These are the two roots of the equation.