"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 ?!] "
