# Convert Cylindrical equation to an equation in spherical coordinates?

So in my homework problem I'm being asked to convert from a cylindrical equation to spherical coordinates and to sketch the graph, The original function is $$z^2+8z=-3r^2-6$$ I'm using the conversions of $r=\rho\sin\phi \quad z=\rho\cos\phi \quad r^2+z^2=\rho^2$ I get stuck and cannot simplify it enough anymore to get $p$ alone. This is what I have so far.$$z^2+3r^2=-8z-6\\\rho^2\cos^2\phi+3\rho^2\sin^2\phi=-8\rho\cos\phi-6\\\rho^2(1-\sin\phi^2)\\\rho^2-\rho^2\sin^2\phi+3\rho^2\sin^2=-8\rho\cos\phi-6\\\rho^2+2\rho^2\sin^2\phi=-8\rho\cos\phi-6$$ I'm not sure where to move on from this if I could get some advice on how to move forward I'd greatly appreciate it.

• I think you got confused and changed $x$ to $z$ when you went from $z^2 + 8x = -3r^2 - 6$ to $z^2 + 3r^2 = -8z -6$. Let us know if you still have problems after correcting that! :) – 2012ssohn Nov 2 '16 at 0:32
• Sorry it was a typo that I made while writing it out, but the questions still stands. – Carlos V Nov 2 '16 at 0:36