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 equationwhich is a

quartic equation in general [11] and its form depends on the medium bidyadic [??].

According to the solution of a

quartic equation in [21], 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 [12] for [k.sub.z], given by