Classify the following quadric surface

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Discussion Overview

The discussion revolves around classifying a specific quadric surface defined by the equation x² - 2xy + 2y² - 2yz + z² = 0. Participants explore methods for manipulating the equation, including completing the square and using hints provided in the thread.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested, Mathematical reasoning

Main Points Raised

  • One participant suggests replacing 2y² with y² + y² as a potential approach to simplify the equation.
  • Another participant proposes using the property of associativity to rewrite the equation as a sum of squares, indicating a possible path forward.
  • Some participants express confusion regarding the terms -2xy and -2yz, questioning whether completing the square is a viable method to address these terms.
  • A later reply asserts that completing the square is unnecessary, claiming that the terms are already squares and referencing the condition under which a² + b² = 0 holds true.

Areas of Agreement / Disagreement

Participants exhibit uncertainty regarding the best method to classify the quadric surface, with some advocating for completing the square while others argue it is not needed. The discussion remains unresolved as no consensus is reached on the approach.

Contextual Notes

There are limitations regarding the assumptions made about the terms in the equation and the definitions of the mathematical properties discussed. The discussion does not resolve the mathematical steps necessary for classification.

scolon94
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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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