Finding the centre and radius by completing the square

The general form of a circle

• A circle is often given in general (expanded) form:

$x^2 + y^2 + Dx + Ey + F = 0$

• In this form the centre and radius are hidden — you cannot read them off directly.
• The plan is to complete the square in $x$ and in $y$ to reach the standard form $(x - h)^2 + (y - k)^2 = r^2$.

How to complete the square for a circle

• Group the $x$-terms together and the $y$-terms together; move the constant to the right.
• For each group, take half the coefficient of the linear term and square it.
• Add that number to both sides (once for $x$, once for $y$).
• Factorise each group into a perfect square bracket.

Reading off the answer

• The completed form is $(x - h)^2 + (y - k)^2 = r^2$.
• Centre $= (h, k)$; radius $= \sqrt{r^2}$.
• If the right-hand side comes out negative, there is no real circle — check for an error.

Why "half then square" works

• Expanding $(x + p)^2 = x^2 + 2px + p^2$ shows the constant is the square of half the $x$-coefficient.
• So to rebuild the bracket from $x^2 + Dx$, you need $p = \dfrac{D}{2}$ and you must add $p^2 = \left(\dfrac{D}{2}\right)^2$.