(x+1)[2x+1]=x+0.5(1)+1.000000001

Solution for (x+1)[2x+1]=x+0.5(1)+1.000000001 equation:

Simplifying
(x + 1)[2x + 1] = x + 0.5(1) + 1.000000001

Reorder the terms:
(1 + x)[2x + 1] = x + 0.5(1) + 1.000000001

Reorder the terms:
(1 + x)[1 + 2x] = x + 0.5(1) + 1.000000001

Multiply (1 + x) * [1 + 2x]
(1[1 + 2x] + x[1 + 2x]) = x + 0.5(1) + 1.000000001
([1 * 1 + 2x * 1] + x[1 + 2x]) = x + 0.5(1) + 1.000000001
([1 + 2x] + x[1 + 2x]) = x + 0.5(1) + 1.000000001
(1 + 2x + [1 * x + 2x * x]) = x + 0.5(1) + 1.000000001
(1 + 2x + [1x + 2x2]) = x + 0.5(1) + 1.000000001

Combine like terms: 2x + 1x = 3x
(1 + 3x + 2x2) = x + 0.5(1) + 1.000000001

Multiply 0.5 * 1
1 + 3x + 2x2 = x + 0.5 + 1.000000001

Reorder the terms:
1 + 3x + 2x2 = 0.5 + 1.000000001 + x

Combine like terms: 0.5 + 1.000000001 = 1.500000001
1 + 3x + 2x2 = 1.500000001 + x

Solving
1 + 3x + 2x2 = 1.500000001 + x

Solving for variable 'x'.

Reorder the terms:
1 + -1.500000001 + 3x + -1x + 2x2 = 1.500000001 + x + -1.500000001 + -1x

Combine like terms: 1 + -1.500000001 = -0.500000001
-0.500000001 + 3x + -1x + 2x2 = 1.500000001 + x + -1.500000001 + -1x

Combine like terms: 3x + -1x = 2x
-0.500000001 + 2x + 2x2 = 1.500000001 + x + -1.500000001 + -1x

Reorder the terms:
-0.500000001 + 2x + 2x2 = 1.500000001 + -1.500000001 + x + -1x

Combine like terms: 1.500000001 + -1.500000001 = 0.000000000
-0.500000001 + 2x + 2x2 = 0.000000000 + x + -1x
-0.500000001 + 2x + 2x2 = x + -1x

Combine like terms: x + -1x = 0
-0.500000001 + 2x + 2x2 = 0

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

Divide each side by '2'.
-0.2500000005 + x + x2 = 0

Move the constant term to the right:

Add '0.2500000005' to each side of the equation.
-0.2500000005 + x + 0.2500000005 + x2 = 0 + 0.2500000005

Reorder the terms:
-0.2500000005 + 0.2500000005 + x + x2 = 0 + 0.2500000005

Combine like terms: -0.2500000005 + 0.2500000005 = 0.0000000000
0.0000000000 + x + x2 = 0 + 0.2500000005
x + x2 = 0 + 0.2500000005

Combine like terms: 0 + 0.2500000005 = 0.2500000005
x + x2 = 0.2500000005

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 = 0.2500000005 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + x2 = 0.2500000005 + 0.25

Combine like terms: 0.2500000005 + 0.25 = 0.5000000005
1.25 + x2 = 0.5000000005

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 0.5000000005

Calculate the square root of the right side: 0.707106782

Break this problem into two subproblems by setting 
(x + 0.5) equal to 0.707106782 and -0.707106782.

Subproblem 1
x + 0.5 = 0.707106782

Simplifying
x + 0.5 = 0.707106782

Reorder the terms:
0.5 + x = 0.707106782

Solving
0.5 + x = 0.707106782

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 = 0.707106782 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 0.707106782 + -0.5
x = 0.707106782 + -0.5

Combine like terms: 0.707106782 + -0.5 = 0.207106782
x = 0.207106782

Simplifying
x = 0.207106782

Subproblem 2
x + 0.5 = -0.707106782

Simplifying
x + 0.5 = -0.707106782

Reorder the terms:
0.5 + x = -0.707106782

Solving
0.5 + x = -0.707106782

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 = -0.707106782 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -0.707106782 + -0.5
x = -0.707106782 + -0.5

Combine like terms: -0.707106782 + -0.5 = -1.207106782
x = -1.207106782

Simplifying
x = -1.207106782

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.207106782, -1.207106782}