x(1+x)=(1-x-x^2)

Solution for x(1+x)=(1-x-x^2) equation:

Simplifying
x(1 + x) = (1 + -1x + -1x2)
(1 * x + x * x) = (1 + -1x + -1x2)
(1x + x2) = (1 + -1x + -1x2)

Remove parenthesis around (1 + -1x + -1x2)
1x + x2 = 1 + -1x + -1x2

Solving
1x + x2 = 1 + -1x + -1x2

Solving for variable 'x'.

Reorder the terms:
-1 + 1x + x + x2 + x2 = 1 + -1x + -1x2 + -1 + x + x2

Combine like terms: 1x + x = 2x
-1 + 2x + x2 + x2 = 1 + -1x + -1x2 + -1 + x + x2

Combine like terms: x2 + x2 = 2x2
-1 + 2x + 2x2 = 1 + -1x + -1x2 + -1 + x + x2

Reorder the terms:
-1 + 2x + 2x2 = 1 + -1 + -1x + x + -1x2 + x2

Combine like terms: 1 + -1 = 0
-1 + 2x + 2x2 = 0 + -1x + x + -1x2 + x2
-1 + 2x + 2x2 = -1x + x + -1x2 + x2

Combine like terms: -1x + x = 0
-1 + 2x + 2x2 = 0 + -1x2 + x2
-1 + 2x + 2x2 = -1x2 + x2

Combine like terms: -1x2 + x2 = 0
-1 + 2x + 2x2 = 0

Begin completing the square.  Divide all terms by
2 the coefficient of the squared term: 

Divide each side by '2'.
-0.5 + x + x2 = 0

Move the constant term to the right:

Add '0.5' to each side of the equation.
-0.5 + x + 0.5 + x2 = 0 + 0.5

Reorder the terms:
-0.5 + 0.5 + x + x2 = 0 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + x + x2 = 0 + 0.5
x + x2 = 0 + 0.5

Combine like terms: 0 + 0.5 = 0.5
x + x2 = 0.5

The x term is x.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
 + 0.25 + x2 = 0.5 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + x2 = 0.5 + 0.25

Combine like terms: 0.5 + 0.25 = 0.75
1.25 + x2 = 0.75

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 0.75

Calculate the square root of the right side: 0.866025404

Break this problem into two subproblems by setting 
(x + 0.5) equal to 0.866025404 and -0.866025404.

Subproblem 1
x + 0.5 = 0.866025404

Simplifying
x + 0.5 = 0.866025404

Reorder the terms:
0.5 + x = 0.866025404

Solving
0.5 + x = 0.866025404

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = 0.866025404 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 0.866025404 + -0.5
x = 0.866025404 + -0.5

Combine like terms: 0.866025404 + -0.5 = 0.366025404
x = 0.366025404

Simplifying
x = 0.366025404

Subproblem 2
x + 0.5 = -0.866025404

Simplifying
x + 0.5 = -0.866025404

Reorder the terms:
0.5 + x = -0.866025404

Solving
0.5 + x = -0.866025404

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = -0.866025404 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -0.866025404 + -0.5
x = -0.866025404 + -0.5

Combine like terms: -0.866025404 + -0.5 = -1.366025404
x = -1.366025404

Simplifying
x = -1.366025404

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.366025404, -1.366025404}