Algebra equation with symbolic

Hello,

I try to solve the 4th order equation with symbolic, but the answer is not numerical numbers, see below:

My code is

 m1 = 100; %kg
 m2 = 10; %kg
 c1 = 1;
 c2 = 1;
 c3 = 1;
 k1 = 100;
 k2 = 100;
 k3 = 100;
 %
 M(1,1) = m1;
 M(2,2) = m2
 %
 C(1,1) = (c1+c2);
 C(1,2) = -c2;
 C(2,1) = -c2;
 C(2,2) = (c2+c3)
 % 
 K(1,1) = (k1+k2);
 K(1,2) = -k2;
 K(2,1) = -k2;
 K(2,2) = (k2+k3)
 %
 a0 = M(1,1)*M(2,2)
 a1 = M(1,1)*C(2,2) + M(2,2)*C(1,1)
 a2 = M(1,1)*K(2,2) + C(1,1)*C(2,2) + M(2,2)*K(1,1) - C(1,2)*C(2,1)
 a3 = C(1,1)*K(2,2) + K(1,1)*C(2,2) - C(1,2)*K(2,1) - C(2,1)*K(1,2)
 a4 = K(1,1)*K(2,2) - K(1,2)*K(2,1)
 %
 syms w
 %
 func = a0*w^4 + a1*w^3 + a2*w^2 + a3*w^1 + a4
 %
 wsol = solve(func,w)

then, the matlab give me a solution which format is root(σ1, z, 1). The sigma is the above function.

How can I get the solution?

Thank you.

Answer 1

[John D'Errico]

4 roots, all of which are complex.

vpa(wsol)
ans =
    - 0.10269696007084728245763107930116 - 4.53087688516932263934459193381i
    - 0.10269696007084728245763107930116 + 4.53087688516932263934459193381i
 - 0.0073030399291527175423689206988387 + 1.208534091963622145606298796162i
 - 0.0073030399291527175423689206988387 - 1.208534091963622145606298796162i

If you plot func, you will see that it never crosses zero.

[Walter Roberson]

You can also double(wsol) instead of vpa(wsol)

Answer 2

[Walter Roberson]

   wsol = simplify(solve(func, w, 'MaxDegree', 4));

This will give you the numeric solutions, such as

((-1)^(1/4)*10^(1/4)*230339930457^(3/4)*(13099491187973 + 159912003^(1/2)*1638000000i)^(1/4)*(- 159912003^(1/2)*1638000000i - 13099491187973)^(1/6)*(2*33315^(1/2)*(- 5456997*30^(1/2)*(- 13099491187973 + 159912003^(1/2)*1638000000i)^(1/2) - 4218680045*10^(1/2)*(8437360090 + 10*(- 13099491187973 + 159912003^(1/2)*1638000000i)^(2/3) - 439697*(- 13099491187973 + 159912003^(1/2)*1638000000i)^(1/3))^(1/2) - 439697*10^(1/2)*(- 13099491187973 + 159912003^(1/2)*1638000000i)^(1/3)*(8437360090 + 10*(- 13099491187973 + 159912003^(1/2)*1638000000i)^(2/3) - 439697*(- 13099491187973 + 159912003^(1/2)*1638000000i)^(1/3))^(1/2) - 5*10^(1/2)*(- 13099491187973 + 159912003^(1/2)*1638000000i)^(2/3)*(8437360090 + 10*(- 13099491187973 + 159912003^(1/2)*1638000000i)^(2/3) - 439697*(- 13099491187973 + 159912003^(1/2)*1638000000i)^(1/3))^(1/2))^(1/2) + 10^(3/4)*2221^(1/2)*(33*(- 13099491187973 + 159912003^(1/2)*1638000000i)^(1/6)*(8437360090 + 10*(- 13099491187973 + 159912003^(1/2)*1638000000i)^(2/3) - 439697*(- 13099491187973 + 159912003^(1/2)*1638000000i)^(1/3))^(1/4) + 3^(1/2)*(8437360090 + 10*(- 13099491187973 + 159912003^(1/2)*1638000000i)^(2/3) - 439697*(- 13099491187973 + 159912003^(1/2)*1638000000i)^(1/3))^(3/4)))*(159912003^(1/2)*8793940000000i + 10*(293032087997 + 159912003^(1/2)*20000000i)*(- 13099491187973 + 159912003^(1/2)*1638000000i)^(1/3) + 133040111*(- 13099491187973 + 159912003^(1/2)*1638000000i)^(2/3) + 16594597703959910)^(1/4)*1i)/7769568131425052256545514000

You should consider whether you actually want the numeric solutions, or if you want approximate results instead, such as the ones John showed.

If what you want is the approximate results then:

wsol_approx = vpasolve(func);