A Novel Necessary and Sufficient Condition for the Positivity of a Binary Quartic Form

Table 2

Degenerate conics in the pencil and properties of three roots of .

Case^a	Degenerate conic	Property of
1	Three real line-pairs
2	One real line-pair + two complex conjugate line-pairs
3	One real line-pair + two complex line-pairs	^b
4	Two real line-pairs
5	One real line-pair + one complex conjugate line-pair
6	One real repeated line + one real line-pair
7	One real repeated line + one complex conjugate line-pair
8	One real line-pair
9	One real repeated line

^aCorresponding to Table 1, ^b are complex conjugate.