Quadratic Forms, Translation
Translation
If a quadratic form includes the terms x2 and 52x,
replace x with u-26.
this is called a translation.
The origin is moved 26 units in the x direction, making the surface easier to analyze.
We can always put it back, once we understand it.
In this case the substitution eliminates the linear term,
leaving u2-676.
The number 676 is subtracted from the constant term that was present in the original quadratic form.
In general, the linear terms can all be eliminated,
simply by moving the origin.
Once this is done, each variable appears squared, or linear, but not both.
And as we saw earlier, several linear variables can be combined into one.
Thus the quadratic form has become a combination of squared variables,
plus a constant, plus at most one linear variable.