Solving an equation with conditional solutions

[peter.a]

f[x_] := x^2 + c;
y = Solve[f'[f[x]] == 1, x, Reals]

output:

{{x -> ConditionalExpression[Root[-1 + 4 c #1 + 4 #1^3 &, 1],
c > -(3/4) || c < -(3/4)]}, {x ->
ConditionalExpression[Root[-1 + 4 c #1 + 4 #1^3 &, 2],
c < -(3/4)]}, {x ->
ConditionalExpression[Root[-1 + 4 c #1 + 4 #1^3 &, 3],
c < -(3/4)]}}

Then i want to put the solutions in terms of c which doesn't work with:
Eliminate[c == -x - x^2 && x == y[[1]], x]

 

[Bill Simpson]

In[1]:= ToRadicals[Root[-1+4 c #1+4 #1^3&,1]]

Out[1]= ((-4*c)/(3^(1/3)*(9 + Sqrt[3]*Sqrt[27 + 64*c^3])^(1/3)) + (9 + Sqrt[3]*Sqrt[27 + 64*c^3])^(1/3)/3^(2/3))/2

but I have NOT carefully verified that this is valid under the ConditionalExpression and even if it is the solutions will still depend on the value of the unknown c.

This still leaves the situation that for some values of c there are 3 solutions, for some there are 2 and for some there are only 1.