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

Enter the roots, separated by commas.### Response: To find the roots of a quadratic equation, I usually try to factor it first.
However, factoring this equation seems difficult, because it involves complex numbers.
I could try to use the quadratic formula, but that might also be messy.
Another option is to complete the square, which might be easier.
To do that, I need to move the constant term to the left side and add a term to both sides that makes the left side a perfect square.
So I get $z^2 - z + \frac{1}{4} = 5 - 5i + \frac{1}{4}.$
The left side is now $(z - \frac{1}{2})^2,$ and the right side is $\frac{21}{4} - 5i.$
Taking the square root of both sides, I get $z - \frac{1}{2} = \pm \sqrt{\frac{21}{4} - 5i}.$
Adding $\frac{1}{2}$ to both sides, I get $z = \frac{1}{2} \pm \sqrt{\frac{21}{4} - 5i}.$
These are the roots of the equation.