  In this paper, by considering the common points of two conics instead of the
roots of the binary quartic form, we propose a novel necessary and sufficient
condition for the positivity of a binary quartic form using the theory of the
pencil of conics. First, we show the degenerate members of the pencil of conics
according to the distinct natures of the common points of two base conics.
Then, the inequalities about the parameters of the degenerate members are
obtained according to the properties of the degenerate conics. Last, from the
inequalities we derive a novel criterion for determining the positivity of a
binary quartic form without the discriminant.
