What geometric object is given by this equation?

Question

What geometric object is given by this equation?

$x^2+y^2+z^2+2xy+2xz+2yz-x-y-z-6=0$

Maple says it's a hyperboloid of one sheet, but is there a way to show it without going the long way by using the principal axis theorem?

Answer 1

Your equation can be transformed to $(x+y+z-3)(x+y+z+2)=0$, so $x+y+z-3=0$ or $x+y+z+2=0$, so it's a union of two parallel planes.