The technique of completing
the square is used to turn a quadratic into the sum of a squared binomial and
a number: (x
– a)^{2} + b.
The center-radius form of the circle equation is in the format (x
– h)^{2} + (y – k)^{2} = r^{2},
with the center being at the point (h,
k) and the radius
being "r".
This form of the equation is helpful, since you can easily find the center
and the radius.

Find the center
and radius of the circle having the following equation: 4x^{2}
+ 4y^{2} – 16x – 24y + 51 = 0.

Here
is the equation they've given you.

Move
the loose number over to the other side.

Group
the x-stuff
together. Group the y-stuff
together.

Whatever
is multiplied on the squared terms (it'll always be the same
number), divide it off from every term.

This
is the complicated step. You'll need space inside your groupings,
because this is where you'll add the squaring term. Take the
x-term
coefficient, multiply it by one-half, square it, and then add
this to both sides of the equation, as shown. Do the same with
the y-term
coefficient. Convert the left side to squared form, and
simplify the right side.

Read
off the answer from the rearranged equation.

The
center is at (h,
k) = (x, y) = (2, 3).
The radius is r
= sqrt(^{ 1}/_{4} ) = ^{1}/_{2}

Completing the square to
find a circle's center and radius always works in this manner. Always
do the steps in this order, and each of your exercises should work out
fine. (Also, if you get in the habit of always working the exercises in
the same manner, you are more likely to remember the procedure on tests.)

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Warning: Don't misinterpret
the final equation. Remember that the circle formula is (x
– h)^{2} + (y – k)^{2} = r^{2}.
If you end up with an equation like (x
+ 4)^{2} + (y + 5)^{2} = 5,
you have to keep straight that h
and k
are subtracted in the center-radius form, so you really have (x
– (–4))^{2} + (y – (–5))^{2} = 5.
That is, the center is at the point (–4,
–5), not at (4,
5). Be careful with
the signs; don't just "read off the answer" without thinking.
Also, remember that the formula says "r^{2}",
not "r",
so the radius in this case is sqrt(5),
not 5.

In the course of the above
procedure, about the only other thing that can be a problem is forgetting
the sign on the step where you multiply by one-half. Warning: If you drop
a negative, you'll get the wrong answer for the coordinates of the center,
so be careful of this. Don't try to do this step in your head: write
it out!

Here's one more example
of how completing the square works for circle equations:

Find the center
and radius of the circle with the following equation: 100x^{2}
+ 100y^{2} – 100x + 240y – 56 = 0.

This
is the given equation.

Move
the loose number over to the other side.

Group
the x-stuff
and y-stuff
together.

Divide
off by whatever is multiplied on the squared terms.

Take
the coefficient on the x-term,
multiply by one-half, square, and add inside the x-stuff
and also to the other side. Do the same with the y-term.

Convert
the left-hand side to squared form, and simplify the right-hand
side.

If
necessary, fiddle with signs and exponents to make your equation
match the circle equation's format.

Read
off the answer.

The
center is at (
^{1}/_{2}, –^{ 6}/_{5} )
and the radius is
^{3}/_{2}.