Chapter 21.2 of Friedlander and Iwaniec's <i>Opera de Cribro</i> begins (essentially) as follows:

Let
$$F(X, Y) = aX^2 + bXY + cY^2 + \alpha X + \beta Y + \gamma \in \mathbb{Z}[X, Y]$$ be irreducible in $\mathbb{Q}[X, Y]$, represent arbitrarily large odd integers, and satisfy $(a, b, c, \alpha, \beta, \gamma) = 1$. We want $F(X, Y)$ to depend essentially on two variables, which may be expressed by the requirement that $\frac{\partial F}{\partial X}, \frac{\partial F}{\partial Y}$ are linearly independent.

We introduce two discriminants:
$$\Delta = b^2 - 4 ac,$$
$$D = a \beta^2 - b \alpha \beta + c \alpha^2 + \Delta \gamma.$$

[end of excerpt]

Wait, they do what? 

F+I go on to state that any $F(X, Y)$ satisfying the above conditions represents infinitely many primes: on the order of $\frac{x}{\log x}$ primes $\leq x$ if $D = 0$ or $\Delta$ is a square, and on the order of $\frac{x}{(\log x)^{3/2}}$ otherwise.

They say only a very little bit about their proof, referring instead to <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa24/aa2451.pdf">this</a> paper of Iwaniec. Obviously these discriminants are important (Iwaniec calls them the "small" and "large" discriminants respectively), but Iwaniec does not explain them in any highbrow fashion; he simply dives into the guts of some computations. Predictably, they are related to affine changes of coordinates (we want to say that $F, F'$ are equivalent if an invertible such transformation changes one into the other), but if the role of $D$ is something simple then this wasn't apparent from a quick look at the paper.

Are these two discriminants classical and familiar? Is there a simple highbrow explanation of what they say about the polynomial $F$? It seems there must be.

(See also <a href="http://mathoverflow.net/questions/55384/primes-represented-by-two-variable-quadratic-polynomials">here</a> for a related MO question.)

Thank you!