Definite and Indefinite Forms

Definition 4.1   A quadratic form with negative discriminant is called definite. A form with positive discriminant is called indefinite.

Let $(a,b,c)$ be a quadratic form. Multiply by $4a$ and complete the square:

$$4a(ax^2+bxy+cy^2) = 4a^2x^2 + 4abxy + 4acy^2$$

$$= (2ax+by)^2 + (4ac-b^2)y^2$$

If $\disc(a,b,c)<0$ then $4ac-b^2=-\disc(a,b,c)>0$, so $ax^2 + bxy+cy^2$ takes only positive or only negative values, depending on the sign of $a$. In this sense, $(a,b,c)$ is very definite about its choice of sign. If $\disc(a,b,c)>0$, then $(2ax+by)^2 + (4ac-b^2)y^2$ takes both positive and negative values, so $(a,b,c)$ does also.

We will consider only definite forms in the next two lectures.