Expand the square, using the F.O.I.L. method:
#3 = x^2 + 8xy + 16y^2 + 3x#
Substitute #rcos(theta) # for x and #rsin(theta)# for y:
#3 = (rcos(theta))^2 + 8(rcos(theta))(rsin(theta)) + 16(rsin(theta))^2 + 3(rcos(theta))#
This can be written as a quadratic in r:
#0 = (cos^2(theta) + 8cos(theta)sin(theta) + 16sin^2(theta))r^2 + 3cos(theta)r - 3#
The coefficient for #r^2# factors into a square:
#0 = (cos(theta) + 4sin(theta))^2r^2 + 3cos(theta)r - 3#
Use the quadratic formula:
#r = (-b +-sqrt(b^2 - 4(a)(c)))/(2a)#
where:
#a = (cos(theta) + 4sin(theta))^2#
#b = 3cos(theta)#
#c = -3#
Also, we must change the #+-# to only #+#, because negative values for r do not make sense.
#r = (-3cos(theta) +sqrt(9cos^2(theta) + 12(cos(theta) + 4sin(theta))^2))/(2(cos(theta) + 4sin(theta))^2)#
