Question

Reduce the equation to one of the standard forms, classify the surface. (No answers without explanations/ work please. I want to understand why)

Answer 1

Sorry, you need an equation: 4y^2 + z^2 - x -16y -4z +20 = 0.

Answer 2

let's complete the squares

-x+ 4(y^2 -4y +4) +(z^2-4z+4)=-20

add 16 and 4 to other side as well
-x+ 4(y-2)^2 +(z-2)^2=-20+16+4

4(y-2)^2 +(z-2)^2=x


In general
-----------
a(y^2)+ b(z^2)=1 tends to represent ellipse


but our equation is not equal to 1 but x

let's to tracing

when z=0

we get
4(y-2)^2 +(0-2)^2=x
4(y-2)^2=x-4

that's parabola, along the z axis it looks like parabola

but on y-z plane, setting x=0

4(y-2)^2 +(0-2)^2=0

this is ellipse,

so it would be elliptical- paraboloid