We need to complete the square for the #x and x^2# terms and then for the #y and y^2# terms.
First, reorder the terms
#-9x^2+72x+4y^2-16y=164#
Factor out a #-9# from the first 2 terms.
*Remember to change the signs.
Factor out a #4# from the last 2 terms.
#-9(x^2-8x)+4(y^2-4y)=164#
Work with the #x# term.
#(-8/2)^2=(-4)^2=16#
Remember that we factor out #-9# so, we have to add #-9*4=-36# to the right side of the equation.
Work with the #y# term.
#(-4/2)^2=(-2)^2=4#
Remember that we factor out #4# so, we have to add #4*4=16# to the right side of the equation.
#-9(x^2-8x+16)+4(y^2-4y+4)=164-144+16#
Factor:
#x^2-8x+16=>(x-4)^2 => # Perfect square trinomial
#y^2-4y+4=>(y-2)^2 =># Perfect square trinomial
#-9(x-4)^2+4(y-2)^2=164-144+16#
#-9(x-4)^2+4(y-2)^2=36#
#(-9(x-4)^2)/36+(4(y-2)^2)/36=36/36#
#-((x-4)^2)/4+((y-2)^2)/9=1#
