find the radius and circle by completing the square

Organization is key to answering this one. First group all the x's and y's together and move the extraneous numbers to the other side. The parenthesis here are to show where we're going to complete the square once the problem is organized correctly.

2x^{2 }+ 8x + (____) + 2y^{2} - 12y + (____) = 24

Then divide through by the coefficients of the two squared terms. You're trying to make this look like the equation for a circle which has the two square binomials set equal to the radius. Anyway, since both coefficients are 2, simply divide the whole thing by two.

x^{2} + 4x + (____) + y^{2} - 6y + (____) = 12

Now look at the x term, 4x. Take half of the coefficient (4/2 = 2) and square it (which brings us back to 4, coincidentally) and add that to both sides. Do the same for the Y term. Write it like this to show clearly what you've done:

x^{2} + 4x + (4) + y^{2} - 6y + (9) = 12 + 4 + 9

Now factor your two perfect square trinomials and add up all the loose numbers on the right.

(x + 2)^{2} + (y - 3)^{2} = 5^{2} (Turned the 25 into 5-squared for the next step. You should do this too)

This looks like:

(x - h)^{2} + (y - k)^{2} = r^{2}

You should be able to pull the correct center coordinates and radius from there.