<s>[INST] Analyzing quadratic forms $x^2-3xy+y^2$, $2xy+yz-3xz$ and $x^2+y^2+2xy-xt+2yt$. Positive, negative or indefinite? 
By means of successive coordinate changes, write each one of the
  quadratic forms below as a sum of terms of the type $\pm u^2$ and
  decide wich ones are positive, negative or indefinite:
$$A(x,y) = x^2-3xy+y^2$$
$$B(x,y,z) = 2xy+yz-3xz$$
$$C(x,y,z,t) = x^2+y^2+2xy-xt+2yt$$

For $A$ I did $A(x,y) = x^2-3xy+y^2 = x^2-2xy+y^2-xy = (x-y)^2-xy$ which is indefinite.
For $B(x,y) = 2xy+yz-3xz$ I remembered that $(x+y+z)^2 = x^2 + 2 x y + y^2 + 2 x z + 2 y z + z^2$ so $2xy+yz-3xz = (x+y+z)^2 -x^2-y^2-z^2-yz+xz$ which won't help anything.
For $C(x,y,z,t)$ there's too much terms, how should I do it?
 [/INST] If you want to work directly with the quadratic expressions, you should just "complete" the square. For $C$, we have
$$ C(x,y,z,t) = x^2 + y^2 + 2xy - xt + 2yt = \left( x + y - \frac{1}{2}t \right)^2 - \frac{1}{4}t^2 + 3yt = \\ \left( x + y - \frac{1}{2}t \right)^2 - \left(\frac{1}{2}t - 3y \right)^2 - 9y^2.$$
I've manipulated the expression to get the terms $x^2,y^2,2xy, -xt$ as part of the expansion of the first square term which resulted in an "error" of the form $-\frac{1}{4}t^2 - yt$. Then I tried to get the terms $-\frac{1}{4}t^2, 3yt$ as part of the expansion of the second square, etc.
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