1. **State the problem:** Find the center and radius of the circle given by the equation $$x^2 + y^2 - 10x + 10y + 25 = 0$$. 2. **Formula and rules:** The general form of a circle's equation is $$x^2 + y^2 + Dx + Ey + F = 0$$. To find the center and radius, we complete the square for both $x$ and $y$ terms. 3. **Group $x$ and $y$ terms:** $$x^2 - 10x + y^2 + 10y = -25$$ 4. **Complete the square:** For $x$: Take half of $-10$ which is $-5$, square it to get $25$. For $y$: Take half of $10$ which is $5$, square it to get $25$. Add $25$ to both sides for $x$ and $25$ for $y$: $$x^2 - 10x + 25 + y^2 + 10y + 25 = -25 + 25 + 25$$ 5. **Rewrite as perfect squares:** $$ (x - 5)^2 + (y + 5)^2 = 25$$ 6. **Identify center and radius:** Center is at $(5, -5)$. Radius is $$\sqrt{25} = 5$$. **Final answer:** Center: $(5, -5)$ Radius: $5$