Question 141518
I think you meant that you want to 'complete the square'


What you are trying to do is develop the equation of a circle in standard form, namely {{{(x-h)^2+(y-k)^2=r^2}}}, an equation of a circle with center at (h,k) and radius r.



So, what do you need to add to {{{x^2+2x}}} so that you will have a perfect square trinomial?  Answer: take the coefficient on the 1st degree term, 2 in this case, divide it by 2, and square the result.  You should get 1 if you did the arithmetic correctly.  So add 1 to both sides of your original equation.  And then the first three terms will be {{{x^2+2x+1}}} which can be factored to {{{(x+1)^2}}}


Now, perform the same process for the y terms, adding the result to both sides of the original equation, and then factoring the y-terms plus the new constant you added.


Since we know that the first part of the result is {{{(x+1)^2}}}, we know that the x-coordinate of the center of the circle must be -1 ({{{(x+1)^2}}} is the same as {{{(x-(-1))^2}}}).  Whatever comes up from doing the y-side will give you the other coordinate.


The sum of the original constant term, 4, the 1 you added to complete the square on the x terms, and whatever you added to complete the square of the y terms is the square of the radius.