This equation is a particular case of a depressed quartic equation
and it can be solved by the Ferrari method, hence reducing it to a depressed cubic equation, and then use Cardano's formulas.
The stability of the endemic equilibrium [E.sub.1] is difficult to prove analytically, because it involves a quartic equation
which depend on the variables [I.sub.m] and [I.sub.f].
Hence, from equation (1) it is soon seen that x satisfies the quartic equation
which is a quartic equation
in general  and its form depends on the medium bidyadic [??].
According to the solution of a quartic equation
in , we can obtain the following solutions:
The solution, in the fifth command, of those four simultaneous and non-linear equations yields results in four sets corresponding to the roots of a quartic equation
. Three of these sets are directly discarded because they contain negative or complex values of pressures; the acceptable solution is displayed in the last line.
Combining Equations (4) and (5) and simplifying, we have an explicit quartic equation
in terms of [T.sub.2]:
Case 2: A quartic equation
with two real roots and one imaginary root of multiplicity 2
Now [X.sub.jkt] includes year fixed effects and a quartic equation
in free throw attempts.
The geometric solution of a quartic equation
by the use of circles and y = [x.sup.2] has been presented.
The quartic equation
A = 0 has roots a = [+ or -](1 + [square root of (1 - [[kappa].sup.2])]v/2[[kappa].sup.2] and [+ or -](1 - [square root of (1 - [[kappa].sup.2])]v/2[[kappa].sup.2] with [a.sub.2] = (2 - [[kappa].sup.2] + 2[square root of (1 - [[kappa].sup.2])]v/2[[kappa].sup.2]/4[[kappa].sup.4] and (2 - [[kappa].sup.2] - 2[square root of (1 - [[kappa].sup.2])][v.sup.2]/4[[kappa].sup.4], respectively.
We will use the Booker quartic equation
derived by Pettis  for [k.sub.z], given by