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# Def/Discriminant of a binary quadratic form

Definition of Discriminant of a binary quadratic form: Suppose that $Q(x,y) = ax^2 + bxy + cy^2$ is a binary quadratic form. The discriminant of $Q$, sometimes written $\Delta(Q)$, is defined by: $$\Delta(Q) = b^2 - 4ac.$$

Alternatively, given the range topograph of $Q$, the discriminant can be computed from any cell in the topograph as pictured, using the formula:

A cell in the topograph

$$\Delta(Q) = (u-v)^2 - ef.$$

The discriminant can be used to classify binary quadratic forms into the following types:

• If $\Delta(Q) < 0$, we say that $Q$ is a definite quadratic form.
• If $\Delta(Q) > 0$, then we say that $Q$ is an indefinite quadratic form.
• If $\Delta(Q) = 0$, then we say that $Q$ is a degenerate quadratic form.

## Logical Connections

This definition logically relies on the following definitions and statements: Def/Binary quadratic form

The following statements and definitions logically rely on the material of this page: State/Periodicity along the river, and State/Solving quadratic Diophantine equations in two variables

To visualize the logical connections between this definition and other items of mathematical knowledge, you can visit any of the following clusters, and click the "Visualize" tab: Clust/Binary quadratic forms