General and standard form of a circle

A circle's equation is sometimes given in an expanded general form. From $x^2 + y^2 - 6x + 4y - 3 = 0$, we convert to the standard form $(x - a)^2 + (y - b)^2 = r^2$, from which the center and radius can be read off.

We complete the square in $x$ and in $y$ separately. Since $x^2 - 6x = (x - 3)^2 - 9$ and $y^2 + 4y = (y + 2)^2 - 4$,

Collecting constants gives $(x - 3)^2 + (y + 2)^2 - 16 = 0$, that is

Comparing with the standard form, the center is $(3, -2)$ and the radius is $\sqrt{16} = 4$.

In general $x^2 + y^2 + \ell x + m y + n = 0$ becomes the standard form of a circle (when the right side is positive) by completing the square in $x$ and $y$. Half of the $x$-coefficient $-6$ is $-3$, and half of the $y$-coefficient $4$ is $2$; these are the center coordinates with signs flipped. The large dots on the graph are the center $(3, -2)$ and the point $(7, -2)$ on the circle, a radius $4$ to its right.