# Question:Depressed quartic equation solution

## Question:Depressed quartic equation solution

I have a depressed quartic equation with the quadratic term also removed. It has at least one positive real root which is the solution that I am looking for. There are three cases to consider. Two of the cases have trivial solutions. Maple gives its usual RootOf solution for the general case. My worksheet is below.

> restart;
> assume(b > 0, g > 0);
Depressed quartic equation with quadratic term also removed.
It has at least one real posivtive root -- the desired root.
> Eq1 := v^4-b*v-g = 0;
4
v  - b v - g = 0
Case 1: b = 0, g > 0
> Eq2 := subs(b = 0, Eq1);
4
v  - g = 0
> Sol2 := solve(Eq2, v);
(1/4)     (1/4)    (1/4)      (1/4)
g     ,  I g     ,   -g     ,   -I g
> v = Sol2[1];
(1/4)
v = g
Case 2: g = 0, b > 0
> Eq3 := subs(g = 0, Eq1);
4
v  - b v = 0
> Sol3 := solve(Eq3, v);
(1/3)     1  (1/3)    1    (1/2)  (1/3)    1  (1/3)   1    (1/2)  (1/3)
0,  b     ,  - - b         + - I 3      b     ,     - - b         - - I 3      b
2               2                          2              2
> v = Sol3[2];
(1/3)
v = b
Case 3: g > 0, b > 0
> Sol1 := solve(Eq1, v);
/   4                 \
RootOf\_Z  - b _Z - g/

I have been playing with the analytical solution to the quartic equation, but I get a royal mess. It seems to me that I should be able to get a "reasonably simple" expression for the positive real root of the simple quartic defined in Eq1 with the given assumptions on b and g, but I don't know enough about the theory of polynomial roots. I am also surprised that Maple doesn't know the analytic solution to the quartic equation.

Maybe I'm being overly optimistic, but will appreciate any help with this problem. Thanks.

Neill S.

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