[Math] Why Mathematica (Reduce) can’t find clear solution for almost trivial inequalities

inequalitymathematica

Suppose we want to solve the inequality $ax^2+bx+c<0$. For simplicity, presume that $a>0$ and $b^2-4ac>0$. In this form, this is almost a trivial problem. Despite that, if we want to solve it with Mathematica (Reduce), version 8, the

Assuming[a > 0 && b^2 - 4*a*c > 0, Reduce[a*x^2 + b*x + c < 0, x, Reals]]

command generates dozens of lines (instead of simply being -(b/(2 a)) - 1/2 Sqrt[(b^2 - 4 a c)/a^2] < x < -(b/(2 a)) + 1/2 Sqrt[(b^2 - 4 a c)/a^2], in which this solution is present somewhere deeply, but it is full of

  • unnecessary case separations (as if it couldn't interpret the $b^2-4ac<0$ condition, and although it was given as assumption, it repeats this in the solution in equivalent forms),
  • impossible conditions (such as $a\leq 0$ which should have been ruled out by the assumptions),
  • branches with clearly contradictory conditions (such as x < -(b/(2 a)) - 1/2 Sqrt[(b^2 - 4 a c)/a^2] || x > -(b/(2 a)) - 1/2 Sqrt[(b^2 - 4 a c)/a^2] at the very beginning).

The usage of

Reduce[{a > 0, b^2 - 4*a*c > 0, a*x^2 + b*x + c < 0}, x, Reals]

and especially

FullSimplify[Reduce[{a > 0, b^2 - 4*a*c > 0, a*x^2 + b*x + c < 0}, x, Reals]]

alleviates the issue a bit (which is itself surprising given of what Assuming should do… or at least what I thought it should do…), but it is still far from what I've expected as a result. (Especially because this last command, although it produces the shortest output, it doesn't even give an explicit solution for $x$.)

Thank you in advance!

Best Answer

As far as I know, Reduce doesn't use the assumptions system, only Refine, Simplify, FullSimplify and FunctionExpand do.

So you could manually put the assumptions into Reduce, then remove them from the result using the simplify mechanism:

$Assumptions = a > 0 && b^2 - 4*a*c > 0;
Reduce[a*x^2 + b*x + c < 0 && $Assumptions, x] // FullSimplify

which yields

output

If you include $b\neq0$ in your assumptions, then you get the simple result that you're looking for: 

$Assumptions = a > 0 && b^2 - 4*a*c > 0 && b != 0;
Reduce[a*x^2 + b*x + c < 0 && $Assumptions, x] // FullSimplify

(* Returns:
(-b - Sqrt[b^2 - 4 a c])/(2 a) < x < (-b + Sqrt[b^2 - 4 a c])/(2 a)
*)
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