Can this equation be solved for x? If yes, how does the solution look? Thank you. Karl.

a*x^4-b*x^2-c*x+d=0

Answer 1

[Askic V]

Things become complicated very easy with high degree polynomials.

First, this is an example for qudratic equation:

syms a b c x

% Define the equation
eqn = a*x^2 + b*x + c  == 0;

% Solve the equation symbolically
sol = solve(eqn, x);

pretty(sol) 

/             2          \
|   b + sqrt(b  - 4 a c) |
| - -------------------- |
|            2 a         |
|                        |
|             2          |
|   b - sqrt(b  - 4 a c) |
| - -------------------- |
\            2 a         /

and this is for 4th order:

syms a b c d x

% Define the equation
eqn = a*x^4-b*x^2-c*x+d  == 0;

% Solve the equation symbolically
% The option specifies the max degree
% of polynomials for which the solver tries to find and return 
% an explicit solutions
sol = solve(eqn, x,'MaxDegree',4);

pretty(sol(1)) % only first solution

                /                                                              /                   3       2          \   \
                |                                                              |                2 b    27 c    72 b d |   |
                |                                                sqrt(6) c sqrt| 3 sqrt(3) #4 - ---- + ----- + ------ | 3 |
                |        1/3                 2                                 |                  3       2       2   |   |
                | b #1 #2    12   d #1 12   b  #1          2/3                 \                 a       a       a    /   |
            sqrt| ------------- - ------- - ----- - 9 #1 #2    - -------------------------------------------------------- |
                |       a            a         2                                             a                            |
     #1         \                             a                                                                           /
- ------- - ---------------------------------------------------------------------------------------------------------------
      1/6                                      /                             1/3 \1/4
  6 #2                                         | 12 d            2/3   6 b #2    |      1/6
                                             6 | ---- + #3 + 9 #2    + --------- |    #2
                                               \   a                       a     /

where

             /                           1/3   \
             | 12 d            2/3   b #2    6 |
   #1 == sqrt| ---- + #3 + 9 #2    + --------- |
             \   a                       a     /

                         3      2
         sqrt(3) #4     b      c     4 b d
   #2 == ---------- - ----- + ---- + -----
             18           3      2       2
                      27 a    2 a     3 a

          2
         b
   #3 == --
          2
         a

             /     4        3       4        3  2        2  2          2   \
             | 27 c    256 d    16 b  d   4 b  c    128 b  d    144 b c  d |
   #4 == sqrt| ----- - ------ - ------- - ------- + --------- + ---------- |
             |    4       3         5         5          4           4     |
             \   a       a         a         a          a           a      /