$-6xy-6xz+3y^2+6yz-2z^2$
I've already tried to factor out some variables, but I am always left with 3 variables again after my transformation. I've tried $(a+b)^2$ and $(a+b+c)^2$, I guess I need some methodology, I really rely on you, guys >.<
2026-03-25 22:02:01.1774476121
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Completion square method quadratic form
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The result of a purely algorithmic approach is $Q^T D Q = H,$ where half the Hessian of your quadratic form is $H,$ then $D$ is diagonal, and $Q$ has determinant $\pm 1.$ I show the matrices first. The method, an algorithm, is discussed in detail at my http://math.stackexchange.com/questions/1388421/reference-for-linear-algebra-books-that-teach-reverse-hermite-method-for-symmetr
===============================================================
? qt
%17 =
[-3/2 3/5 -1]
[3/2 -1 0]
[-1 0 0]
? d
%18 =
[-2 0 0]
[0 15/2 0]
[0 0 9/5]
? q
%19 =
[-3/2 3/2 -1]
[3/5 -1 0]
[-1 0 0]
? qt * d * q
%20 =
[0 0 -3]
[0 3 3]
[-3 3 -2]
?
===============================================================
The whole thing:
parisize = 4000000, primelimit = 500509
? h = [ 0,0,-3; 0,3,3; -3,3,-2]
%1 =
[0 0 -3]
[0 3 3]
[-3 3 -2]
? ht = mattranspose(h)
%2 =
[0 0 -3]
[0 3 3]
[-3 3 -2]
? h - ht
%3 =
[0 0 0]
[0 0 0]
[0 0 0]
? p1 = [ 0,0,1; 0,1,0; 1,0,0]
%4 =
[0 0 1]
[0 1 0]
[1 0 0]
? h1 = p1 * h * p1
%5 =
[-2 3 -3]
[3 3 0]
[-3 0 0]
? p2 = [ 1, 3/2, -3/2; 0,1,0; 0,0,1]
%6 =
[1 3/2 -3/2]
[0 1 0]
[0 0 1]
? p2t = mattranspose(p2)
%7 =
[1 0 0]
[3/2 1 0]
[-3/2 0 1]
? h2 = p2t * h1 * p2
%8 =
[-2 0 0]
[0 15/2 -9/2]
[0 -9/2 9/2]
? p3 = [ 1,0,0; 0,1, 9/15; 0,0,1]
%9 =
[1 0 0]
[0 1 3/5]
[0 0 1]
? p3t = mattranspose(p3)
%10 =
[1 0 0]
[0 1 0]
[0 3/5 1]
? h3 = p3t * h2 * p3
%11 =
[-2 0 0]
[0 15/2 0]
[0 0 9/5]
? d = h3
%12 =
[-2 0 0]
[0 15/2 0]
[0 0 9/5]
? p = p1 * p2 * p3
%13 =
[0 0 1]
[0 1 3/5]
[1 3/2 -3/5]
? matdet(p)
%14 = -1
? q = matadjoint(p)
%15 =
[-3/2 3/2 -1]
[3/5 -1 0]
[-1 0 0]
? qt = mattranspose(q)
%16 =
[-3/2 3/5 -1]
[3/2 -1 0]
[-1 0 0]
? qt
%17 =
[-3/2 3/5 -1]
[3/2 -1 0]
[-1 0 0]
? d
%18 =
[-2 0 0]
[0 15/2 0]
[0 0 9/5]
? q
%19 =
[-3/2 3/2 -1]
[3/5 -1 0]
[-1 0 0]
? qt * d * q
%20 =
[0 0 -3]
[0 3 3]
[-3 3 -2]
?
===============================================================
Complete the square to deal with all the $z$'s \begin{eqnarray*} -2(z-\frac{3}{2}y +\frac{3}{2}x)^2 + ... \end{eqnarray*} Now complete the square for the $x$'s \begin{eqnarray*} -2(z-\frac{3}{2}y +\frac{3}{2}x)^2 + \frac{1}{2} (3 x -5y)^2 -5y^2. \end{eqnarray*}