Simplying linear equation to get quartic in q with using Maple and using Descarte’s rule of sign

[ra_forever8]

Using the maple I am trying to get quardic in q from this big linear equation. Then use Descarte’s rule of signs to determine the number of positive roots.
\begin{equation}
\frac{\gamma*q*P_Q}{k_p*(1-q)*P_C} = \frac{I*\alpha}{k_f+k_d+\frac{k_n*\lambda_b*\gamma*q*P_Q}{\lambda_b*\gamma*q*P_Q+k_p*\lambda_r*(1-q)^2}+k_p*(1-q)}
\end{equation}
Values of parameters are given below:
$I=1200$
$k_f = 6.7*10.^7$
$k_d = 6.03*10.^8$
$k_n = 2.92*10.^9$
$k_p = 4.94*10.^9$
$\alpha = 1.14437*10.^(-3)$
$\lambda_b = 0.87e-2$
$\lambda_r = 835$
$\gamma = 2.74$
$P_C = 3*10.^(11)$
$P_Q = 2.87*10.^(10)$

=>
I tried the code in maple to get quartic in q but DOES NOT WORK.

         II := 1200: 
k_f := 6.7*10.^7: 
k_d := 6.03*10.^8: 
k_n := 2.92*10.^9: 
k_p := 4.94*10.^9: 
alpha := 1.14437*10.^(-3): 
lambda_b := 0.87e-2:
 lambda_r := 835:
 ggamma := 2.74:
 P_C := 3*10.^11: 
P_Q := 2.87*10.^10:
        
         eq := ggamma*q*P_Q/(k_p*(1-q)*P_C) = II*alpha/(k_f+k_d+k_n*lambda_b*ggamma*q*P_Q/(lambda_b*ggamma*q*P_Q+k_p*lambda_r*(1-q)^2)+k_p*(1-q)):
simply(eq, q);

My lecturer want me to manipulate the equation and get a quartic in q before substituting the values of parameters into the equation. After that,use Descarte’s rule of signs to determine the number of positive roots. Then write Q=1-q to get second quartic in Q and repeat rule of signs to determine number of steady states of q less than 1. And do the substition of parameters if necessary.
Now, its kind of hard for me what he wants because to get quartic in q first from the equation is hard to do by hand , so i have to use in maple which is not working then use Descarte’s rule of signs.

 

[mmwave]

Thank you for your post. I understand that you are struggling to manipulate the given equation and obtain a quartic in q. I would like to offer some suggestions that may help you in your task.

Firstly, it is important to understand that manipulating equations and obtaining specific forms can be challenging, especially when dealing with complex equations like the one provided. It is not uncommon for the process to take multiple steps and involve the use of mathematical software like Maple.

To begin, I suggest starting with the original equation and simplifying it as much as possible. This may involve combining terms, factoring, and using algebraic techniques. Once you have simplified the equation, you can start to manipulate it to obtain a quartic in q.

One approach you can take is to multiply both sides of the equation by the denominators and then expand the resulting expression. This will result in a quartic equation in q. You can then use Maple to solve for q and obtain the four roots.

Once you have the quartic in q, you can use Descarte's rule of signs to determine the number of positive roots. This rule states that the number of positive roots of a polynomial equation is either equal to the number of sign changes in the coefficients or less than that by an even number. For example, if the polynomial has three sign changes, then it can have either three or one positive roots.

After determining the number of positive roots, you can substitute q=1-Q and repeat the process to obtain a second quartic equation in Q. This will allow you to determine the number of steady states of q less than 1.

I hope this helps you with your task. Remember to take your time and use Maple or other mathematical software to assist you in the process. Good luck!