#2x^2 +y^2 +3x +4y = 0#
Step 1. Group the #x# and #y# terms separately.
#(2x^2 + 3x )+ (y^2 + 4y) = 0#
Step 2. Complete the square for the #x# terms.
#2x^2 + 3x = 0#
#2(x^2 +3/2x) = 0#
#(3/2)^2/4 = (9/4)/4 = 9/16#
#2(x^2 + 3/2x + 9/16 - 9/16) = 0#
#2((x+3/4)^2 -9/16) = 0#
#2(x+3/4)^2 - 9/8 = 0#
#2(x+3/4)^2 =9/8#
Step 3. Complete the square for the #y# terms.
#y^2 + 4y = 0#
#4^2/4 = 16/4 = 4#
#y^2 + 4y +4 -4= 0#
#(y+2)^2 – 4 = 0#
#(y+2)^2 = 4#
Step 4. Recombine the #x# and #y# terms.
#2(x+3/4)^2 = 9/8#
#(y+2)^2 = 4#
#bar(2(x+3/4)^2 + (y+2)^2 = 9/8 +4#
#2(x+3/4)^2 + (y+2)^2 = 9/8 +32/8#
#2(x+3/4)^2 + (y+2)^2 = 41/8#
Check:
#2(x+3/4)^2 + (y+2)^2 -41/8#
#= 2(x^2 + 3/2x + 9/16) + (y^2 +2y+4) - 41/8#
#= 2x^2 +3x +9/8 +y^2 + 2x +4 -41/8#
#= 2x^2 +y^2+3x + 2y +9/8 +4 - 41/8#
#2x^2 +y^2+3x + 2y + 41/8 -41/8 =0#
#2x^2 +y^2+3x + 2y = 0#
