Question

Use the equation

Answer 1

This is the equation for a circle! Start by grouping all the x terms together and do the same with the y terms so you get this: (x^2 - 14x) + (y^2 - 18y) + 105 = 0
Now move the 105 over to the other side to get it out of the way (it is part of the radius so it will be moved over there eventually anyway). Now we have:
(x^2 - 14x) +(y^2 - 18y) = -105.  To complete the square on the x terms, take half the linear term (the 14) which is 7 and square it to get 49.  That's what's added in to complete the square on the x's.  But since you added it on the left, you also have to add it to the -105 on the right:
(x^2 - 14x + 49) + (y^2 - 18y) = -105 + 49 so far.  Now do the same to the y terms by taking half of the linear term (the 18) and squaring it to get 81. That's what's added in on the left so it also has to be added in on the right:
(x^2 -14x + 49) + (y^2 - 18y +81) = -105 + 49 + 81.  Simplifying all of this mess reduces to: (x-7)^2 + (y-9)^2 = 25 which is the equation of a circle with a center of (7, 9) and a radius of 5. Those are so much fun!