Substitute #rcos(theta)# for x and #rsin(theta)# for y:
#3(rcos(theta))(rsin(theta)) = 2(rcos(theta))^2 - 3rcos(theta) + 2(rsin(theta))^2#
Factor out r whenever possible:
#3r^2(cos(theta))(sin(theta)) = 2r^2cos^2(theta) - 3rcos(theta) + 2r^2sin^2(theta)#
Swap the last two terms:
#3r^2(cos(theta))(sin(theta)) = 2r^2cos^2(theta) + 2r^2sin^2(theta) - 3rcos(theta) #
Combine the #r^2# terms on the right:
#3r^2(cos(theta))(sin(theta)) = 2r^2(cos^2(theta) + sin^2(theta)) - 3rcos(theta) #
Substitute 1 for #cos^2(theta) + sin^2(theta):#
#3r^2(cos(theta))(sin(theta)) = 2r^2 - 3rcos(theta) #
Combine like terms:
#r^2(3cos(theta))(sin(theta)) - 2) = - 3rcos(theta) #
Move everything to the left side:
#r^2(3cos(theta))(sin(theta)) - 2) + 3rcos(theta) = 0 #
Divide both sides by #(3cos(theta))(sin(theta)) - 2)#
#r^2 + r((3cos(theta))/(3(cos(theta))(sin(theta)) - 2)) = 0 #
Divide both sides by r:
#r + ((3cos(theta))/(3(cos(theta))(sin(theta)) - 2)) = 0 #
Solve for r:
#r = -((3cos(theta))/(3(cos(theta))(sin(theta)) - 2))#
