Classify the following quadric surface

[scolon94]

x²-2xy +2y²-2yz+z²=0
hint: 2y²=y²+y²
I thought of replacing 2y^2. But I'm not sure exactly what to do.
Thank you.

 

[MarkFL]

Use the given hint allow with the property of associativity to write:

\(\displaystyle \left(x^2-2xy+y^2 \right)+\left(y^2-2yx+z^2 \right)=0\)

Now do you see how to proceed?

 

[scolon94]

Use the given hint allow with the property of associativity to write:

\(\displaystyle \left(x^2-2xy+y^2 \right)+\left(y^2-2yx+z^2 \right)=0\)

Now do you see how to proceed?

I'm still confused because of the -2xy and -2yz. Is it possible to complete the square?

 

[MarkFL]

I'm still confused because of the -2xy and -2yz. Is it possible to complete the square?

What is the expansion of $(a-b)^2$?

 

[HallsofIvy]

The point is that you don't need to "complete the square"- those are already squares.

Also [tex]a^2+ b^2= 0[/tex] if and only if a= b= 0.