Canonical form for $x^2+y^2+z^2+xy+xz+yz$, using Gauss

Question

I need to find a canonical form for the following equation, throughout the Gauss method.

$$x^2+y^2+z^2+xy+xz+yz$$

And I'm stuck at this point, because even if I continue to creeate $(a+b+c)^2$ it seems to return same result:

$$(x+y+z)^2-xy-xz-yz$$

Answer 1

The idea is to complete the square:

$$x^2+y^2+z^2+xy+xz+yz=\left(x+\frac{1}{2}(y+z)\right)^2+y^2+z^2-\frac{1}{4}(y+z)^2+yz.\\ =\left(x+\frac{1}{2}(y+z)\right)^2+\frac 3 4 y^2+\frac 3 4 z^2+\frac 1 2yz=\left(x+\frac{1}{2}(y+z)\right)^2+\frac 3 4\left(y+\frac1 3z\right)^2+\frac {2} {3}z^2.$$

Answer 2

Hint: Reduce the equation $$ \left( ax+ay \right) ^{2}+ \left( bx+z \right) ^{2}+ \left( cy+c z \right) ^ {2}-{x}^{2}-{y}^{2}-{z}^{2}-xy-xz-yz=0 $$ to system of equations.

After solution you will get $$ \left( \frac{\sqrt {2}}{2}x+ \frac{\sqrt {2}}{2}y \right) ^{2}+ \left( \frac{\sqrt {2}}{2}y+ \frac{\sqrt {2}}{2}z \right) ^{2}+ \left( \frac{\sqrt {2}}{2}x+ \frac{\sqrt {2}}{2}z \right) ^{2}=\\={x}^{2}+{y}^{2}+{z}^{2}+xy+xz+yz $$