Question 202800
To find (h, k) and R we need to manipulate the equation so it is in the standard form for circles: {{{(x - h)^2 + (y - k)^2 = R^2}}}. This requires that we have the sum of two perfect squares on the left. So we need to "complete the squares" on the left side so we can rewrite them as (x - h)^2 and (y - k)^2.<br>
To complete the squares:<ol><li>Gather all the variable terms on one side (and all the constant terms on the other side:
Subtract 15 from both sides
{{{3x^2 + 3y^2 - 24x - 12y = 15}}}</li><li>Change subtractions, if any, into additions so you can rearrange the terms (using the Commutative Property) and get the x terms and the y terms together:
{{{3x^2 + 3y^2 + (-24x) + (-12y) = 15}}}
{{{3x^2 + (-24x) + 3y^2 + (-12y) = 15}}}</li><li>Since the coefficients of the x^2 and Y^2 terms are the same, divide both sides by this coefficient.
{{{x^2 + (-8x) + y^2 + (-4y) = 5}}}</li><li>Now comes the tricky part. Complete the square for each variable by<ol><li>Figuring out what 1/2 of the coefficient of the first power term is.
For the x's: 1/2 of -8 is -4
For the y's: 1/2 of -4 is -2</li><li>Squaring this "1/2 of the coefficient.
For the x's' (-4)^2 = 16
For the y's' (-2)^2 = 4</li><li>Add these squares to both sides of the equation. On the left side, add the squares after the terms you used to calculate them.
{{{x^2 + (-8x) + 16 + y^2 + (-4y) + 4 = 5 + 16 + 4}}}
{{{x^2 + (-8x) + 16 + y^2 + (-4y) + 4 = 25}}}</li><li>Since we want (x - h)^2 and (y - k)^2, we'll rewrite the equation with subtractions.
{{{x^2 -8x + 16 + y^2 -4y + 4 = 25}}}</li><li>Both the first three terms on the left and the last three terms on the left fit the perfect square pattern: {{{a^2 - 2ab + b^2 = (a - b)^2}}}
So we now rewrite each set of three terms as the perfect squares they are:
{{{(x - 4)^2 + (y - 2)^2 = 25}}}</li></ol></li></ol>The only thing left to do is to write the right side as a perfect square:
{{{(x - 4)^2 + (y - 2)^2 = 5^2}}}
Now that we have transformed
{{{3x^2 + 3y^2 - 24x - 12y = 15}}}
into 
{{{(x - 4)^2 + (y - 2)^2 = 5^2}}}
which is in the standard form (for circles) of (x - h)^2 + (y - k)^2 = r^2 we can read the center: (4, 2) and the radius: 5 and use these to graph the circle.