The Discriminant of a Binary Quadratic Form

The Discriminant of a Binary Quadratic Form

Definition: The Discriminant of the binary quadratic form $f(x, y) = ax^2 + bxy + cy^2$ is the number $d = b^2 - 4ac$.

If $f(x, y) = 4x^2 - 3xy + y^2$ then the discriminant of $f$ is:

(1)
\begin{align} \quad d = (-3)^2 - 4(4)(1) = 9 - 16 = -7 \end{align}

The following proposition tells us when we can factor a binary quadratic form into a product of linear factors.

Proposition 1: Let $f(x, y) = ax^2 + bxy + cy^2$ be a binary quadratic form and let $d$ be the discriminant of $f$.
a) If $d$ is a perfect square ($d = 0, 1, 4, 9, ...$) then $f(x, y)$ can be factored as a product of two linear factors. In particular, if $d = 0$ then $f(x, y)$ is the square of a linear factor and if $d = 1, 4, 9, ...$ then $f(x, y)$ is the product of two distinct linear factors.
b) If $d$ is not a perfect square then $f(x, y)$ cannot be factored as a product of two linear factors with integer coefficients.
Proposition 2: If $f(x, y) = ax^2 + bxy + cy^2$ is a binary quadratic form and $d$ is not a perfect square then the only integer solution to $f(x, y) = 0$ is the solution $(x, y) = (0, 0)$.
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