Elimination of Cross Terms in Equation of Surface By Completion of Square Example

Suppose we have a quadratic equation  
\[\sum_{ij, \: i \leq j}^n a_{ij}x_ix_j\]
  that describes a surface in  
. We can eliminated the cross terms  
\[Ax_ix_j, \: i \neq j\]
  by completing the square.

We can complete the square for  


Add and subtract  


Now set  
\[y_1=x_1-x_2, \: y_2=2x_2, \: y_3 =x_2+2x_3\]
  to give

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