## 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
$\mathbb{R}^n$
. We can eliminated the cross terms
$Ax_ix_j, \: i \neq j$
by completing the square.
Example:
$x_1^2-2x_1x_2+4x_2x_3-2x_2^2+4x_3^2=1$

We can complete the square for
$x_1^2-2x_1x_2=(x_1-x_2)^2-x_2^2$

Hence
$(x_1-x_2)^2-x_2^2+4x_2x_3-2x_2^2+4x_3^2=(x_1-x_2)^2-3x_2^2+4x_2x_3+4x_3^2=1$

$x_2^2$
$(x_1-x_2)^2-4x_2^2+x_2^2+4x_2x_3+4x_3^2=(x_1-x_2)^2-4x_2^2+(x_2+2x_3)^2=1$
$y_1=x_1-x_2, \: y_2=2x_2, \: y_3 =x_2+2x_3$
$y_1^2-y_2^2+y_3^2=1$