How Can Quadratic Surface Equations Be Simplified Efficiently?

[JamesGoh]

Homework Statement

Write the following quardatic surface equation as a sum of multiples of
squares of independent linear functions

$x^{2}+4y^{2}+56z^{2}+2xy+4xz+28yz$

Homework Equations

The Attempt at a Solution

Please see attachment.

nb. there is no answer provided by the tute, but from all the previous exercises, it should
be in the form of something like $(ax + by -cz)^{2} + (dx + ey -fz)^{2} + g^{2}$


My problem is that I have no effective and quick way to solve these sorts of problems using the complete the squares method.

I would have to spend hours trying to find a combination that will make the equation balance out

 

[Nino MMP]

, which is time consuming and not always successful. In the end I just got lucky this time and found a combination that worked. Can anyone suggest a better way to approach these problems? x^2 + 4y^2 + 56z^2 + 2xy + 4xz + 28yz = (x + 2y - 7z)^2 + (2x + 7z)^2 + 49y^2