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The graph of the equation $$ x^2-2 x+y^2+4 y = 4 $$ is a circle. Find the center and radius of that circle.

We can start by writing out our equation like so: $ x^2 - 2x +(-1)^2 - (-1)^2 + y^2 + 4y + 2^2 - 2^2 = 4$

From there we can clean it up a bit $ x^2 - 2x +1 -1 + y^2 + 4y + 4 - 4 = 4$

And then do some factoring $(x-1)^2-1 + (y+2)^2 - 4 = 4$

And add the constant terms over $(x-1)^2 + (y+2)^2 = 9$

And voila this equation tells us that the circle is centered at (1,-2) with a radius of 3.

## Comments

We can start by writing out our equation like so:

$ x^2 - 2x +(-1)^2 - (-1)^2 + y^2 + 4y + 2^2 - 2^2 = 4$

From there we can clean it up a bit

$ x^2 - 2x +1 -1 + y^2 + 4y + 4 - 4 = 4$

And then do some factoring

$(x-1)^2-1 + (y+2)^2 - 4 = 4$

And add the constant terms over

$(x-1)^2 + (y+2)^2 = 9$

And voila this equation tells us that the circle is centered at (1,-2) with a radius of 3.