Putting an equation in quadratic form

[miahmad]

Could someone please explain how to go about putting an equation into quadratic form.
e.g:
Q(x,y,z)=7x^2-2y^2-40z^2-14xz+20yz.

I know it equals 7(x-z)^2 -2(y-5z)^2 +3z^2. dnt know how to get there though.

 

[miahmad]

Dnt wrry figured it out.

 

[CompuChip]

What you want to do is first make a square which gets all the x terms right, then a square which gets all the y terms right, but without touching the x. Then in the end, you can use the z² term to balance the z's.

So, for example, with an x you have 7x² and -14xz.
If you write c (x - a)², you will get terms with c x² and 2 x a c, and stuff with a², which you can cancel by -a² terms, if you need. So you can reproduce these two terms by taking c = 7, a = -z.

Then you do the same with the y, choosing numbers c and a such that you exactly get the right y² and yz terms.

Then, you multiply out everything you have so far. If you did it right, all the terms will match, except possibly those with only z². Since z² is itself a square, you can easily fix this by adding the appropriate amount of z² at the end, and you're done.