1. **State the problem:** We are given the equation of a circle: $$x^2 + y^2 + 2x - 2y = 0$$ and need to find its radius. 2. **Rewrite the equation:** The general form of a circle is $$(x - h)^2 + (y - k)^2 = r^2$$ where $(h,k)$ is the center and $r$ is the radius. 3. **Complete the square:** Group $x$ and $y$ terms: $$x^2 + 2x + y^2 - 2y = 0$$ 4. Complete the square for $x$: $$x^2 + 2x = (x + 1)^2 - 1$$ 5. Complete the square for $y$: $$y^2 - 2y = (y - 1)^2 - 1$$ 6. Substitute back: $$(x + 1)^2 - 1 + (y - 1)^2 - 1 = 0$$ 7. Simplify: $$(x + 1)^2 + (y - 1)^2 - 2 = 0$$ 8. Add 2 to both sides: $$(x + 1)^2 + (y - 1)^2 = 2$$ 9. **Identify radius:** The radius squared is 2, so radius is $$r = \sqrt{2}$$ **Final answer:** The radius of the circle is $\sqrt{2}$.