Different quadratic forms

Question

I don't have a math background and am familiar with only the standard form of the quadratic formula, $y=ax^2 + bx + c$

What is the proper name for this formula I've just seen also referred to as a quadratic form?:

$f(w)=\frac{1}{2}w^TAw-b^Tw$

Answer 1

The quadratic formula is an analytic solution to the question of finding the roots of a polynomial of degree $2$ over a field.

A quadratic form is a homogeneous polynomial of degree two in a number of variables. You can also formulate this abstractly as a function $q:V\to F$ where $V$ is an $F$ vector space, and $q$ has special properties.

All of this is completely explained in the wiki pages

https://en.wikipedia.org/wiki/Quadratic_formula

https://en.wikipedia.org/wiki/Quadratic_form

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Given any symmetric matrix $A$, $w\mapsto w^\top Aw$ defines a quadratic form. The $-b^\top w$ term makes it not a true quadratic form, but as I search around I find it called an extended quadratic form in other places.