Discriminant of an Equation - Discriminant Test

Here is the Discriminant Test.

DISCRIMINANT TEST

If we ignore the degenerate cases, the graph of a second degree equation is:

A parabola if the discriminant is zero. An ellipse if the discriminant is negative. A hyperbola if the discriminant is positive.

For example, the equation

xy - 1 = 0

has positive discriminant

1² - 4 · 0 = 1,

and its graph is a hyperbola. The equation

2x² + xy + y² - 1 = 0

has negative discriminant

1² - 4 · 2 · 1 = -7,

and its graph is an ellipse.

The degenerate graphs that can arise are: two straight lines, one straight line, one point, and the empty graph. The Discriminant Test alone does not tell whether or not the graph is degenerate. However, a degenerate case can usually be recognized when one tries to sketch the graph. For the remainder of this section we shall ignore the degenerate cases.