# solving system of polynomial equations over reals using newton method

I have a set of polynomial equations and I want to find one of its real solutions close to some point, and I need only one solution. Here is an example:

This is the list of equations and variables:

```
Equations = [x_0*x_1*x_2*x_3 - x_0*x_1 - x_0*x_2 - x_0*x_3 - x_1*x_2 - x_1*x_3 + 2*x_0 + 2*x_1 - 448, -x_0*x_1*x_2 - x_0*x_1*x_3 - x_0*x_2*x_3 - x_1*x_2*x_3 + 3*x_0 +
3*x_1 + 2*x_2 + 2*x_3 + 452, x_0*x_1 + x_0*x_2 + x_0*x_3 + x_1*x_2 + x_1*x_3 + x_2*x_3 - 159, -x_0 - x_1 - x_2 - x_3 + 21]
Variables = [x_0, x_1, x_2, x_3]
```

If I ask Sage to solve this

```
S = solve(Equations,Variables)
```

it returns a bunch of solutions. But in some cases it doesn't give me any real solutions. I can prove that the above set of equations has a real solution close to `[2,4,7,8]`

. Is there any way that I can perform an algorithm like the Newton's method with the start point `[2,4,7,8]`

, and find that real solution?