Circles in expanded form

Quick reminder. A circle with centre $(a, b)$ and radius $r$ has equation

$(x - a)^2 + (y - b)^2 = r^2.$

Why? A point $(x, y)$ is on the circle exactly when its distance from the centre is $r$. The horizontal gap is $x - a$ and the vertical gap is $y - b$, so Pythagoras gives (gap across)² + (gap up)² = $r^2$. This centre–radius form lets you read the centre and radius straight off. Note the signs: $(x - 3)$ means the centre has $x$-coordinate $+3$.

Now expand $(x - 3)^2 + (y + 2)^2 = 25$ and tidy up:

$x^2 - 6x + 9 + y^2 + 4y + 4 = 25 \;\Rightarrow\; x^2 + y^2 - 6x + 4y - 12 = 0.$

Same circle, but now the centre and radius are scrambled. This is the expanded form. Your job is to unscramble it back to centre–radius form: complete the square in $x$, and separately in $y$.

Remember the completing-the-square rule: $x^2 + kx = \left(x + \tfrac{k}{2}\right)^2 - \left(\tfrac{k}{2}\right)^2$. Halve the coefficient of $x$ for the bracket, then subtract that half squared.