Obtain the standard form

$4x^2\+2y^2-z^2-8 x\+4 y\+4 z\+2\=0$$\stackrel{\text{manipulate equation}}{\to }$$2{\left(y\+1\right)}^2\+4{\left(x-1\right)}^2\={\left(z-2\right)}^2$

Obtain the equivalent of the surfaces in Figures 3.3.11(a, b)

$4x^2\+2y^2-z^2-8 x\+4 y\+4 z\+2\=0$$\to$

Define $f$ so that the graph of $f\=0$ is a quadric surface

$f≔4x^2\+2y^2-z^2-8 x\+4 y\+4 z\+2\:$

Complete the square and put $f$ into standard form

$\mathrm{Student}:-\mathrm{Precalculus}:-\mathrm{CompleteSquare}\left(f\=0\,\left[x\,y\,z\right]\right)$

$\+{\left(z-2\right)}^2$

$\mathrm{rhs}\left(\right)\=\mathrm{lhs}\left(\right)$

Obtain the equivalent of the surfaces drawn in Figures 3.3.11(a, b)

$\mathrm{plots}:-\mathrm{implicitplot3d}\left(f\=0\,x\=-1..3\,y\=-4..2\,z\=-2..6\,\mathrm{style}\=\mathrm{surfacecontour}\,\mathrm{grid}\=\left[25\,25\,25\right]\,\mathrm{scaling}\=\mathrm{constrained}\,\mathrm{axes}\=\mathrm{frame}\,\mathrm{tickmarks}\=\left[4\,6\,8\right]\,\mathrm{orientation}\=\left[-50\,60\,0\right]\right)$