I am trying to solve the following system of $6$ quadratic equations in $6$ variables:

```
system = 
  { 1/144 (Sqrt[3] x (Sqrt[5] + 1) + 2Sqrt[3] (t1 + t2 - 1) + 4 Sqrt[3] t1 
   - 2Sqrt[3] t2 + 2 Sqrt[3] (x - 1))^2 + 1/16 (x (Sqrt[5] + 1) - 2t1 - 4t2 - 2x + 4)^2
  + 1/9 (x (Sqrt[3] - 3Sqrt[1/2 Sqrt[5] + 7/6]) + 3(x - 1) Sqrt[1/2 Sqrt[5] + 7/6])^2 == s,
   1/144 (Sqrt[3] x (Sqrt[5] + 1) + 2Sqrt[3] (x - 1) + 2Sqrt[3] (Sqrt[5] + 1))^2 
  + 1/16 (x (Sqrt[5] + 1) - 2x + 2)^2 + 1/9 (x (Sqrt[3] - 3Sqrt[1/2 Sqrt[5] + 7/6]) 
  + 3*(x - 1) Sqrt[1/2 Sqrt[5] + 7/6] - Sqrt[3] + 3 Sqrt[1/2 Sqrt[5] + 7/6])^2 == s, 
   1/36 (Sqrt[3] (t1 + t2 - 1) + 2 Sqrt[3] t1 - Sqrt[3] t2 - Sqrt[3] (Sqrt[5] + 1))^2 
  + 1/4 (t1 + 2t2 - 1)^2 + 1/9 (Sqrt[3] - 3Sqrt[1/2 Sqrt[5] + 7/6])^2 == s, 
    1/144 (Sqrt[3] (u1 + u2 - 1) (Sqrt[5] + 1) - 2Sqrt[3] (t1 + t2 - 1)
  - 4Sqrt[3] t1 + 2 Sqrt[3] t2 + 2Sqrt[3]u1 - 4Sqrt[3] u2)^2 + 
    1/16 ((u1 + u2 - 1) (Sqrt[5] + 1) - 2t1 - 4t2 - 2u1 + 2)^2 + 
    1/9 ((u1 + u2 - 1)(Sqrt[3] - 3Sqrt[1/2 Sqrt[5] + 7/6]) 
  + 3u1 Sqrt[1/2 Sqrt[5] + 7/6] + 3u2 Sqrt[1/2 Sqrt[5] + 7/6])^2 == s, 
    1/144 (Sqrt[3] (u1 + u2 - 1) (Sqrt[5] + 1) + Sqrt[3] x (Sqrt[5] + 1) 
  + 2Sqrt[3] u1 - 4 Sqrt[3] u2 + 2Sqrt[3] (x - 1))^2 + 
    1/16 ((u1 + u2 - 1)(Sqrt[5] + 1) - x (Sqrt[5] + 1) - 2u1 + 2x - 2)^2
  + 1/9 ((u1 + u2 - 1)(Sqrt[3] - 3Sqrt[1/2 Sqrt[5] + 7/6]) 
  + x (Sqrt[3] - 3Sqrt[1/2 Sqrt[5] + 7/6]) + 3u1 Sqrt[1/2 Sqrt[5] + 7/6]
  + 3u2 Sqrt[1/2 Sqrt[5] + 7/6] + 3(x - 1)Sqrt[1/2 Sqrt[5] + 7/6])^2 == s, 
    1/144 (Sqrt[3] (u1 + u2 - 1)(Sqrt[5] + 1) + 2Sqrt[3] u1
  - 4Sqrt[3] u2 - 2Sqrt[3](Sqrt[5] + 1))^2 + 1/16 ((u1 + u2 - 1)(Sqrt[5] + 1) - 2u1)^2 
  + 1/9 ((u1 + u2 - 1)(Sqrt[3] - 3Sqrt[1/2 Sqrt[5] + 7/6]) 
  + 3u1 Sqrt[1/2 Sqrt[5] + 7/6] + 3u2 Sqrt[1/2 Sqrt[5] + 7/6] +
      Sqrt[3] - 3Sqrt[1/2 Sqrt[5] + 7/6])^2 == s};

```

Unfortunately *Mathematica* does not give a solution:

```
Solve[ system, {s, t1, t2, u1, u2, x}]

```

It ran for about an hour and it didn't finish.  

 Numerically the system can be solved with `NSolve`.  

I am looking for the solution, where all the variables (`{s, t1, t2, u1, u2, x}`) are positve.  

Numerically it should be this one:

```
{s -> 1.8156, t1 -> 0.290762, t2 -> 0.352453, 
 u1 -> 0.332044,  u2 -> 0.0729072, x -> 0.645495}

```

Is there a way to obtain the exact solutions?  

Things I tried that didn't work:

* setting the domain to `Reals`. (Then even `Nsolve` doesn't find a solution)
* adding an extra inequality `s > 0`.
* eliminating the variable `s` by hand, which gives a system of $5$ quadratic equations with $5$ variables.