Obtain the standard form

$3 x^2- 6 x- y\+ 1\=0$

$3x^2-6x-y\+1\=0$

$\stackrel{\text{complete square}}{\=}$

$3{\left(x-1\right)}^2-y-2\=0$

$\stackrel{\text{add y+2 to both sides}}{\to }$

$3{\left(x-1\right)}^2\=y\+2$

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

$f≔3 x^2- 6 x- y\+ 1\:$

Complete the square and put $f$ into standard form

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

$-\left(-y-2\right)$

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

Obtain the equivalent the surface in Figures 3.3.4(a, b)

$\mathrm{plots}:-\mathrm{implicitplot3d}\left(f\=0\,x\=0..2\,y\=-2..1\,z\=0..5\,\mathrm{scaling}\=\mathrm{constrained}\,\mathrm{style}\=\mathrm{surfacecontour}\,\mathrm{axes}\=\mathrm{frame}\,\mathrm{orientation}\=\left[50\,75\,0\right]\,\mathrm{tickmarks}\=\left[3\,4\,5\right]\right)$