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?

Just use Sylvester's criterion. For example: $$A(x,y) = \begin{pmatrix} x & y \end{pmatrix} \begin{pmatrix} 1 & -3/2 \\ -3/2 & 1 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}.$$The minors of that middle matrix are $1$ and $1 - 9/4 <0 $, so $A$ is indefinite. Writing the expression as the sum of square terms amount to diagonalizing that matrix, but you don't need this to classify the quadratic form. You'll also have $$B = \begin{pmatrix} 0 & 1 & -3/2 \\ 1 & 0 & 1/2 \\ -3/2 & 1 & 0
\end{pmatrix}, \quad C = \begin{pmatrix}1 & 1 & 0 & -1/2 \\ 1 & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ -1/2 & 1 & 0 & 0 \end{pmatrix}.$$
