Classify the following quadric surface

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SUMMARY

The discussion focuses on classifying the quadric surface defined by the equation x² - 2xy + 2y² - 2yz + z² = 0. Participants suggest using the hint 2y² = y² + y² and the property of associativity to rewrite the equation as (x² - 2xy + y²) + (y² - 2yz + z²) = 0. The conversation emphasizes that the terms already represent squares, negating the need for completing the square. The conclusion drawn is that a² + b² = 0 holds true only when both a and b equal zero, indicating the surface is degenerate.

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

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

Now do you see how to proceed?
 
MarkFL said:
Use the given hint allow with the property of associativity to write:

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

Now do you see how to proceed?

I'm still confused because of the -2xy and -2yz. Is it possible to complete the square?
 
scolon94 said:
I'm still confused because of the -2xy and -2yz. Is it possible to complete the square?

What is the expansion of $(a-b)^2$?
 
The point is that you don't need to "complete the square"- those are already squares.

Also a^2+ b^2= 0 if and only if a= b= 0.
 

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