I need help interpreting this passage from page 35 of the book Diophantine Analysis by Robert D Carmichael.
"Thus the problem of solving Eq. (1) is reduced to that of solving Eq. (4) for $u$ and $v$ and choosing one of those values for which x and y have the desired characteristic. But the problem of solving Eq. (4) is identical with that of the representation of a given integer by means of a binary quadratic form. The plan of this book does not permit the detailed development of this latter subject."
Equation 1 is $ax^2+2bxy+cy^2+2dx+2ey+f=0$ and equation 4 is $au^2+2buv+cv^2=m$. $u,v,$ and $m$ are defined elsewhere if that matters, but what I need to know is what that bold part is referring to; and that last sentence gives me little reason to believe I can find context in the book to make sense of it.
This is the main part of the book I'm interested in, but I do intend to go through most of it. It wasn't easy to find a book on the subject and I just checked it out from the library today.
A binary quadratic form is a homogeneous polynomial of degree 2 in two variables, $$q(x,y) = ax^2 + bxy + cy^2.$$
When $a,b,c\in\mathbb{Z}$, we can view $q$ as a function $q\colon\mathbb{Z}\times\mathbb{Z}\to\mathbb{Z}$, a common question asks for a description of the elements of the image. For instance, there is the classic problem of determining which integers can be written as the sum of two squares; this can be viewed as asking for the image of the binary quadratic form $q(x,y) = x^2+y^2$ (that is, $a=c=1$, $b=0$). The classic terminology is that an integer $m$ is "represented" by the binary quadratic form $q$ if and only if there exist integers (sometimes positive integers) $r$ and $s$ such that $q(r,s)=m$.
This subject has deep roots; they go back to Fermat (who described all primes that can be represented by $q(x,y)=x^2+y^2$, $q(x,y) = x^2+2y^2$, and $q(x,y)=x^2+3y^2$), Euler (who proved Fermat's assertions and then attempted to extend them to other quadratic forms $q(x,y) = x^2+dy^2$ with $d$ squarefree), Lagrange, and Gauss (who dedicates the entire Part V of his Disquisitiones Arithmeticae to the study of these functions). They also reach into the East: Brahmagupta's identity, for example, shows that if $m$ and $n$ can both be represented by the quadratic form $q(x,y) = x^2+cy^2$, then so can $mn$ (that is, the set of representable integers is closed under products).
What the book is saying is that, via a change of variable, solving the general two-variable quadratic diophantine equation $$ax^2 + bxy + cy^2 + dx + ey + f = 0\tag{1}$$ is equivalent to the question of determining when a certain integer $m$ is representable by a particular binary quadratic form (where $m$ and the quadratic form depends on $a$, $b$, $c$, $d$, $e$, and $f$); in the sense that solving the diophantine equation $(1)$ yields a representation of $m$; and a representation of $m$ gives a solution to $(1)$.
The subject of quadratic forms is rich and interesting; whole books have been written on the subject by great minds; the Disquisitiones spends more time on them than on everything else put together. Whether an integer can be represented by a binary quadratic form is intimately connected with the arithmetic of quadratic fields. There is a beautiful [Theorem of Conway and Schneeberger] that says that and integral quadratic form with integer matrix that represents the positive integers between 1 and 15 will, in fact, represent all integers. And much more.
But because it is rich and interesting, it is too complicated to get into in any detail in a book that is not devoted to them. Hence, your book is saying that this is a little too hard to get into in detail.