The first thing to do is get #x^2# and #y^2# by themselves without a coefficient. Luckily, all of the terms in this equation are multiples of 4, so I can simplify this by dividing by 4:
#x^2 + y^2 + 4x - 3y = 0#
Now I need to group the like terms and add numbers to complete the squares.
#(x^2 + 4x) + (y^2 - 3x) = 0#
Just a quick trick - when you have the #x^2# and the #x# term and you need to find the 'unit' term, just take half of the #x# term and square it.
#(x^2 + 4x + 2^2) + (y^2 - 3x + (-1.5)^2) = 2^2 + (-1.5)^2#
#(x^2 + 4x + 4) + (y^2 - 3x + 2.25) = 4 + 2.25#
Now I can "unsquare" the polynomials.
#(x+2)^2 + (y-1.5)^2 = 6.25#
And since we know a circle with center (a, b) and radius 'r' has the equation #(x - a)^2 + (y - b)^2 = r^2#, we can say that this circle has a center of (-2, 1.5), and a radius of #sqrt6.25#, or 2.5.
