# How to find roots of a quartic function with unknown constant coefficients?

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Hrinay on 27 Nov 2023
Edited: Walter Roberson on 29 Nov 2023
The polynomial is:
x^4 + 10.1ax^3 + 1.81x^2 + 9ax + 0.81 = 0
where a is an unkown constant coefficient.
I have to find the corresponding x values to further solve an ODE.

James Tursa on 28 Nov 2023
Edited: James Tursa on 28 Nov 2023
You can look here:
But the nature of the roots is going to depend on the value of a. Do you know anything at all about a? What is the context of the overall problem you are solving? What is the ODE?
Hrinay on 28 Nov 2023
No it just represents a dampening coefficient for a spring mass system. The ode models the movement of a building. Essentially a fbd with 2 masses, 2 springs, and 2 dampeners. I get two odes that model the forces on the mass block. I am trying to solve the simultaneous set of ODE's using cramers rule than MUC because the odes all have constant coefficients of which we have a value for every constant except the dampening coefficients C1 and C2.
The FBD:

Torsten on 28 Nov 2023
Edited: Torsten on 28 Nov 2023
syms x a
p = x^4 + 10.1*a*x^3 + 1.81*x^2 + 9*a*x + 0.81;
sol = solve(p==0,'MaxDegree',4)
sol =
vpa(subs(sol,a,1)) % E.g. for a = 1
ans =
Walter Roberson on 29 Nov 2023
Edited: Walter Roberson on 29 Nov 2023
The roots get complicated and fragile
filename = fullfile(tempdir, 'sol.m');
syms x a
p = x^4 + 10.1*a*x^3 + 1.81*x^2 + 9*a*x + 0.81;
sol = solve(p==0);
solF = matlabFunction(sol, 'file', filename)
solF = function_handle with value:
@sol
A = linspace(-1/2,1/2,200);
Y = cell2mat(arrayfun(solF, A, 'uniform', 0));
figure();
plot(A, real(Y(1,:)), A, real(Y(2,:)), A, real(Y(3,:)), A, real(Y(4,:)));
legend({'1 real', '2 real', '3 real', '4 real'})
figure();
plot(A, imag(Y(1,:)), A, imag(Y(2,:)), A, imag(Y(3,:)), A, imag(Y(4,:)));
legend({'1 imag', '2 imag', '3 imag', '4 imag'})