Take the 2-minute tour ×
Mathematica Stack Exchange is a question and answer site for users of Mathematica. It's 100% free, no registration required.

I want to find all values of the paramaters $m$ for which the equation $$t^4 + (2 m - 1) t^2 - 18 m - 72=0$$ has four real distinct solutions less than 2. Put $t = x^2$, I tried

Clear[f];
f[x_] := x^2 + (2 m - 1) x - 18 m - 72;
d := Discriminant[f[x], x];
sol = Solve[f[x] == 0, x];
x1 = x /. sol[[1]];
x2 = x /. sol[[2]];
Reduce[{d > 0, x1 > 0, x2 > 0, x2 < 4}, m]

-6 < m < -4

How can I reduce my code?

share|improve this question
add comment

1 Answer

up vote 2 down vote accepted

You can notice that t==-3, t==3 are always solutions, so not all four solutions will be less than 2.

Factor[t^4 + (2 m - 1) t^2 - 18 m - 72]
(* (-3 + t) (3 + t) (8 + 2 m + t^2) *)

So the problem reduces to the second degree polynomial :

Reduce[Less @@ Join[Solve[8 + 2 m + t^2 == 0, t][[All, 1, 2]], {2}], m]
(* -6 < m < -4 *)

Update :

With[{sols = Solve[t^4 + 2 (m + 1) t^2 + m + 7 == 0, t][[All, 1, 2]]},
 Reduce[Join[Unequal @@@ Subsets[sols, {2}], # < 2 & /@ sols], m, Reals]
]
(* -(31/9) < m < -3 *)
share|improve this answer
    
How about with equation t^4 + 2 (m + 1) t^2 + m + 7==0? –  minthao_2011 Jan 15 at 7:53
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.