Taking another approach, using Select to pick the root(s) which satisfies kf - kb * x > 0

The functions:

F[x_, k_, kf_, kb_] := (2*k/3)*x^3 - (kf - kb*x)^2
allRoots[k_, kf_, kb_] := 
 x /. Solve[F[x, a, b, c] == 0, x, Reals] /. {a -> k, b -> kf, c -> kb} // ToRadicals // ComplexExpand // Simplify
specRoot[k_, kf_, kb_] := Select[allRoots[k, kf, kb], kf - kb*# > 0 &]

Trying it out on some cases:

rootA = specRoot[10, Exp[-1], Exp[1]]
rootA // N
(* {0.103642} *)

rootB = specRoot[1, 2, 3]
rootB // N
(* {0.554337} *)

rootC = specRoot[1, 3, 2]
rootC // N
(* {1.0566} *)

The last case has a single real root.

But the root 0.740044... does not, in fact, satisfy the condition kf - kb*x > 0. It has a too big value. I think the condition is satisfied only by the smallest of the three real positive roots that occur for these data (I previously stated erroneously that it was the middle root).

Leslaw

Yes, you are right. And I was wrong. Since the solutions involves cube roots the explicit imaginary unit (in the last two) can of course sometimes be cancelled.

k=10, kf=Exp[-1] and kb=Exp[1] satisfies the condition:

0 < k < (2 kb^3)/(9 kf) && kb > 0 && kf > 0

And that condition occurs in all three solutions given by Solve

Leslaw, all three solutions are depending on conditional expressions. And they are, as you remarked, mostly "parrot like" repetitions of the input conditions. But the last two of the three has an imaginary unit in the nonconditional part. So no way those two solutions could convert to a real expression.

F[x_, k_, kf_, kb_] := (2*k/3)*x^3 - (kf - kb*x)^2
sols = Solve[{F[x, k, kf, kb] == 0, k > 0, kf > 0, kb > 0}, x, Reals] // ToRadicals;
Column@sols

But there is a new condition imposed to the first solution:

k > (2 kb^3)/(9 kf) && kb > 0 && kf > 0 || 0 < k < (2 kb^3)/(9 kf) && kb > 0 && kf > 0

Give it a try:

x /. First[sols] /. {k -> 4, kb -> 3, kf -> 2}

Hi, thanks, I tried this, but the result still raises questions. Firstly, the result contains conditional expressions which involve conditions such as kf>0, kb<0, etc. Why these conditions are still present, given the fact that it was assumed in the Solve command that k>0, kf>0 and kb>0 ? It looks as if these assumptions were ignored.

Secondly, how do you know that one root is real and the other two complex? All three formulae involve a square root which may produce the imaginary unit if the expression under the root is negative. There is evidence that often there are three real roots, which is easy to check by plotting the polynomial for k=10,kf=Exp[-1],kb=Exp[1]. Isn't there any way to transform/simplify the results returned by Solve to obtain evidently real-valued formulae for these real roots, together with the conditions specifying when they happen to be real?

Ultimately, what I need is a formula for the (supposedly single, but how to prove this?) real root satisfying an additional condition kf-kb*x >0. If I include this condition in the above command:

Solve[{F[x, k, kf, kb] == 0, k > 0, kf > 0, kb > 0, kf-kb*x>0}, x, Reals] // ToRadicals

I obtain indeed one formula for the root x. However, at least for k=10, kf=Exp[-1], kb=Exp[1] it is a wrong formula, because for such returned x there is kf-kb*x < 0.

Leslaw