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?

All you need to do is to plug random numbers in. If an expression yields both positive and negative numbers, it is indefinite. 
For example, for $B(x,y,z)$, when $x=1$, $y=3$, $z=2$, the expression becomes $2(1)(3) + (3)(2) - 3(1)(3) = 3$, which is positive. When $x=10$, $y=1$, $z=100$, the expression becomes $2(10)(1) + (1)(100) - 3(10)(100) = -2880$ which is negative. Therefore, $2xy + yz - 3xz$ is indefinite.
Try using this method to evaluate question c). Although this method is outside the rules, you can use this method to check any mistakes in $u^2$.
