solve function does not give me value of root

my code is simple:

syms x y a b c
a=0.5;
b=0.0092;
c=0.04;
y=x^2+1/x^3;
solve(a*x^2+b*y+c)

matlab gives:

ans =

   RootOf(z^5 + (100*z^3)/1273 + 23/1273, z)[1]
   RootOf(z^5 + (100*z^3)/1273 + 23/1273, z)[2]
   RootOf(z^5 + (100*z^3)/1273 + 23/1273, z)[3]
   RootOf(z^5 + (100*z^3)/1273 + 23/1273, z)[4]
   RootOf(z^5 + (100*z^3)/1273 + 23/1273, z)[5]

BUT, when I change y to y=x^2+1

matlab gives:

ans =

   -(156579^(1/2)*1i)/1273
    (156579^(1/2)*1i)/1273

WHY is that? I dont see there is so much difference. hope someone could help me, THanks!!

[Roger Stafford]

The difference between "y=x^2+1/x^3" and "y=x^2+1" is that the first leads to a fifth degree polynomial equation and the second to a quadratic equation. We all learned how to solve quadratic equations in high school, but mathematicians have shown that "In algebra, the Abel–Ruffini theorem (also known as Abel's impossibility theorem) states that there is no general algebraic solution—that is, solution in radicals—to polynomial equations of degree five or higher with arbitrary coefficients". See

https://en.wikipedia.org/wiki/Abel–Ruffini_theorem

[ZA niceguy]

wow thank you! That really cleared up my confusion!

Answer 1

[Star Strider]

Use the vpasolve function instead:

syms x y a b c
a=0.5;
b=0.0092;
c=0.04;
y=x^2+1/x^3;
x_sol = vpasolve(a*x^2+b*y+c)

x_sol =

                                           -0.4157571755823635878339736950704
   - 0.12544667013145313389016347683479 - 0.46115251800229378352756949075526i
   - 0.12544667013145313389016347683479 + 0.46115251800229378352756949075526i
        0.33332525792263492780715032437 - 0.28135841861289745219925388215346i
        0.33332525792263492780715032437 + 0.28135841861289745219925388215346i

[ZA niceguy]

thanks! this one helps. Still curious about why sometimes solve function does not give root? is it because the equation too complex that a theoretical correct root is not available?

[Star Strider]

My pleasure.

It’s not that it’s too complex, it’s that analytic solutions to quintic and higher polynomials simply don’t exist. See Roger Stafford’s Comment to your original Question, with respect to the Abel–Ruffini theorem, for a comprehensive discussion.

[Walter Roberson]

general analytic solutions do not exist. Some quintics and higher can be factored to lower order polynomials.