Extract the x values that make the two functions values equal

Question

I have two functions for the x variable. I want to extract x values that make the functions equal.

(1)

a + b - a b + c x - b c x + d x - a d x- a e x + a b e x- b f x + a b f x- c d x + a d e x + b c f x - a b e f x

(2)

a + b - a b + c x - b c x + d x - a d x - a e x + a b e x - b f x + 
 a b f x - c d x^2 + a d e x^2 + b c f x^2 - a b e f x^2

where $x\in[0,1]$ equal to each other (i.e. (1)=(2) )

Is it possible to make a program according to the structure of the functions in order to be equal according to the values of x?

A program that extracts x values in the form of a table that makes the functions equal. where $x\in[0,1]$

I need a program that prints 20 $x_i$ values in the form of a table with equal functions

variables = exp // Variables // Sort;

    exp = -Subscript[x, 3] Subscript[x, 4] Subscript[x, 5] Subscript[x, 6]
   Subscript[x, 7] + 
 Subscript[x, 1] Subscript[x, 3] Subscript[x, 4] Subscript[x, 5]
   Subscript[x, 6] Subscript[x, 7] + 
 Subscript[x, 2] Subscript[x, 3] Subscript[x, 4] Subscript[x, 5]
   Subscript[x, 6] Subscript[x, 7] - 
 Subscript[x, 1] Subscript[x, 2] Subscript[x, 3] Subscript[x, 4]
   Subscript[x, 5] Subscript[x, 6] Subscript[x, 7];

exper = -\!\(\*SubsuperscriptBox[\(x\), \(3\), \(2\)]\) Subscript[x, 4]
   Subscript[x, 5] Subscript[x, 6] Subscript[x, 7] + Subscript[x, 1] 
\!\(\*SubsuperscriptBox[\(x\), \(3\), \(2\)]\) Subscript[x, 4]
   Subscript[x, 5] Subscript[x, 6] Subscript[x, 7] + Subscript[x, 2] 
\!\(\*SubsuperscriptBox[\(x\), \(3\), \(2\)]\) Subscript[x, 4]
   Subscript[x, 5] Subscript[x, 6] Subscript[x, 7] - 
 Subscript[x, 1] Subscript[x, 2] 
\!\(\*SubsuperscriptBox[\(x\), \(3\), \(2\)]\) Subscript[x, 4]
   Subscript[x, 5] Subscript[x, 6] Subscript[x, 7]

;

 Module[{expValue, experValue, varValues}, 
     With[{nbrOfResults = 20}, 
        Table[Catch[
           Do[If[0.8 <= (expValue = 
                Round[exp /. 
                  Thread[variables -> (varValues = 
                      Round[RandomReal[{0, 1}, Length[variables]], .1])], 
                 0.01]) < 1, 
             Throw[{NumberForm[#, {3, 1}] & /@ varValues, 
                NumberForm[expValue, {4, 5}], 
                NumberForm[
                 exper /. Thread[variables -> varValues], {4, 5}]} // 
               Flatten]], 10000]], {nbrOfResults}] // 
         SortBy[#, #[[-2]] &] &] // 
       Prepend[#, {variables, "exp", "exper"} // Flatten] & // 
      Grid[#, Frame -> All] &]

This program prints values of functions if they are larger than 0.5

I want to print values of variables if the functions are equal

Answer 1

Your particular equations factorize:

eqn1 = a + b - a b + c x - b c x + d x - a d x - a e x + a b e x - 
   b f x + a b f x - c d x + a d e x + b c f x - a b e f x ;

eqn2 = a + b - a b + c x - b c x + d x - a d x - a e x + a b e x - 
   b f x + a b f x - c d x^2 + a d e x^2 + b c f x^2 - a b e f x^2;

Solve[eqn1 == eqn2, x]

(* {{x -> 0}, {x -> 1}} *)

So either $x=0$ or $x=1$ (for all parameter values). The equations actually factorize, so you can see that these are the only solutions.

Simplify[eqn1 - eqn2]

(* (c - a e) (d - b f) (-1 + x) x == 0 *)