What is Quadratic Form in Math?

By Xah Lee. Date:

recently, finally learned what Quadratic form is.

quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,

4*x^2 + 2*x*y - 3*y^2

is a quadratic form in the variables x and y.

(and Homogeneous polynomial is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; the sum of the exponents in each term is always 5.)

you hear about quadratic form all the time. Though, i don't know why is it significant. Is there something about them that makes them fundamental? (as in “given structure X, it is the {least, most} something?”) Or is it something happens to be a calculational convenience?