# "Recall that the" \ z"-intercepts of any subset" \ S sube RR^3 \ "are the" #
# "points where" \ S \ "intersects the" \ z"-axis." #
# "Recall further that the" \ z"-axis consists of all points of" \ RR^3 \ "of the" #
# "form" \ (0, 0, z), "where" \ z \in RR. \ "I.e., all points of" \ RR^3, "where" \ \ x = 0 \ \ "and" \ \ y=0. #
# "So we are given the surface:" #
# \qquad \qquad \qquad \qquad \qquad 2 x^2 - z^2 - xy -8 yz + y - z - 2 \ = \ 0. \qquad \qquad \qquad \quad (1) #
# "So to find the" \ z"-intercepts of this surface, we look for all" #
# "points of the surface where:" \ \ x = 0 \ \ "and" \ \ y=0. #
# "Thus, we solve eqn. (1), with:" \ \ x = 0 \ \ "and" \ \ y = 0. #
# \qquad \qquad \qquad \qquad \quad \ 2 x^2 - z^2 - xy -8 yz + y - z - 2 \ = \ 0. #
# "Let" \ \ x = 0 \ \ "and" \ \ y=0; "and continue solving for" \ z: #
# \qquad \qquad ( 2 \cdot 0^2 ) - z^2 - ( 0 \cdot 0 ) - ( 8 \cdot 0 \cdot z ) + 0 - z - 2 \ = \ 0 #
# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad
- z^2 - z - 2 \ = \ 0 #
# \qquad \qquad \qquad \qquad \qquad \quad \ - (- z^2 - z - 2 ) \ = \ - ( 0 ) #
# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ z^2 + z + 2 \ = \ 0. #
# "The quadratic expression is not factorable, so we solve by the" #
# "Quadratic Formula:" #
# \qquad \qquad z \ = \ { - b \pm \sqrt{ b^2 - 4 a c} } / { 2 a } \ = \ { - 1 \pm \sqrt{ 1^2 - 4 \cdot 1 \cdot 2} } / { 2\cdot 1 } #
# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ = \ { - 1 \pm \sqrt{ 1^2 - 4 \cdot 1 \cdot 2 } } / { 2\cdot 1 } #
# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ = \ { - 1 \pm \sqrt{ -7 } } / { 2 } #
# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ = \ { - 1 \pm i \sqrt{ 7 } } / { 2 } \quad. #
# :. \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad z \ = \ { - 1 \pm i \sqrt{ 7 } } / { 2 } \quad. #
# :. \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ "no real solutions for" \ z. #
# :. \qquad \qquad \qquad \qquad \qquad \ "this suface has no" \ z"-intercepts." #
# "This is our solution." #
# "[This may be the problem they meant, but I wonder if the" #
# "question was copied correctly ?!] " #
