Things become complicated very easy with high degree polynomials.
First, this is an example for qudratic equation:
eqn = a*x^2 + b*x + c == 0;
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:
eqn = a*x^4-b*x^2-c*x+d == 0;
sol = solve(eqn, x,'MaxDegree',4);
pretty(sol(1))
/ / 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 /
