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 $(x, y) = (0, 0)$.

Example 1

Let $f(x, y) = 3x^2 + 5xy + 2y^2$. Verify proposition 1 (a).

We have that the discriminant of $f$ is:

(2)

\begin{align} \quad d = b^2 - 4ac = (5)^2 - 4(3)(2) = 1 \end{align}

Since $1$ is a perfect square, proposition 1 tells us that $f(x, y)$ can be factored as a product of two distinct linear factors. To accomplish this, we complete the square:

(3)

\begin{align} \quad f(x, y) &= 3x^2 + 3xy + 2xy + 2y^2 \\ &=3x(x + y) + 2y(x + y) \\ &= (3x + 2y)(x + y) \end{align}