# Expressing a quadratic form in canonical form using Lagrange

Tags:
1. Dec 19, 2016

### Apothem

• Moved from technical forums, so no template
Problem:

q=x1x2+x1x3+x2x3

in canonical form using Lagrange's Method/Algorithm

Attempt:
Not really applicable in this case due to the nature of my question

Using the change of variables:
x1=y1+y2
x2=y1-y2
x3=y3
Indeed you get q'=(y1+y3)2-(y2)2-(y3)2

I can see that when you substitute it in it does indeed reduce it to canonical form however my question is as follows:
How am I meant to arrive at the above change of variables? I have seen a few cases of completing the square, but to my knowledge we can't use that here. EDIT: We can

Any help is greatly appreciated!!

EDIT:
I think I've been able to deduce it....
x1x2+x1x3+x2x3=(x3)2+x3(x2+x1)-(x3)2 and from there I completed the square, the only thing I'm still a bit uncertain on is that I worked this out the other way i.e. I worked out y in terms of x and not x in terms of y....

Last edited: Dec 19, 2016