what is the center of a circle and the radius of 4x^2+4x+4y^2-16y-127=0

Alan has an interesting approach.......mine is a little different........but it should still work out the same...

Let's divide the original equation by 4....so we have

x^2 + x + y^2 - 4y - 127/4 =0          add  127/4 to both sides

x^2 + x + y^2 - 4y = 127/4                now, complete the sqaure on x and y

(x^2 + x + 1/4) + (y^2 - 4y + 4) =  127/4 + 1/4 + 4         simplifying, we have

(x + 1/2)^2  + (y - 2)^2  = 144/4

(x + 1/2)^2 + (y - 2 )^2 = 36

So. the center is (-1/2, 2) and the radius is 6

Always good to have a couple of different approaches (even better when they give the same answers!).

Yep....unlike the "keycode" problem..... in which the number of different answers might actually be approaching the same limit as the number of actual keycodes......(LOL!!!)