1. The problem statement, all variables and given/known data Find the coordinates of the point P on the surface of the paraboloid z=6x2+6y2-(35/6) where the normal line to the surface passes through the point (25/6, (25√22)/6, -4). Note that a graphing calculator may be used to solve the resulting cubic equation. 2. Relevant equations Gradient of the paraboloid ∇ƒ 3. The attempt at a solution This question has been stumping me all day as I don't know how to go about finding the point, even though it seems like it should be a piece of cake. I already understand from class that the gradient gets me the normal line slope which is ∇ƒ = <12x, 12y, -1> (where ƒ is the paraboloid function) and from this and the given point can form the parametric version of the normal line like so: n = (25/6) + 50t, (25/6)√22 + (50√22)t, -4 - t It's after this part where I'm lost, it seems from here I need to find t to finally solve for the 3 coordinates which from what I remember of linear algebra can be done by inputting the normal vector into the paraboloid function, yet the note about having to solve a cubic equation leaves me confused as doing it this way only ends up with an ugly quadratic. If anyone can point me in the right direction it would be greatly appreciated as I'm pretty sure I'm missing something important here.