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Classify the following quadric surface

scolon94

New member
Feb 5, 2014
8
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

Administrator
Staff member
Feb 24, 2012
13,775
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

New member
Feb 5, 2014
8
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

Administrator
Staff member
Feb 24, 2012
13,775
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

Well-known member
MHB Math Helper
Jan 29, 2012
1,151
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.