Circle Center Radius 8564E8

1. **State the problem:** We are given the equation $x^2 - 4x + y^2 + 2y = 31$ and need to find the center and radius of the circle it represents. 2. **Rewrite the equation:** Group $x$ and $y$ terms: $$x^2 - 4x + y^2 + 2y = 31$$ 3. **Complete the square:** For $x$ terms: $x^2 - 4x$. Take half of $-4$ which is $-2$, square it: $(-2)^2 = 4$. For $y$ terms: $y^2 + 2y$. Take half of $2$ which is $1$, square it: $1^2 = 1$. Add these squares to both sides to keep equality: $$x^2 - 4x + 4 + y^2 + 2y + 1 = 31 + 4 + 1$$ 4. **Rewrite as perfect squares:** $$ (x - 2)^2 + (y + 1)^2 = 36 $$ 5. **Identify center and radius:** Center is at $(2, -1)$. Radius is $ adius = \\sqrt{36} = 6$. **Final answer:** Center: $(2, -1)$ Radius: $6$