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

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


Simplifying
(2x + 1) * 2 = 1 + (x + 1)(x + -1)

Reorder the terms:
(1 + 2x) * 2 = 1 + (x + 1)(x + -1)

Reorder the terms for easier multiplication:
2(1 + 2x) = 1 + (x + 1)(x + -1)
(1 * 2 + 2x * 2) = 1 + (x + 1)(x + -1)
(2 + 4x) = 1 + (x + 1)(x + -1)

Reorder the terms:
2 + 4x = 1 + (1 + x)(x + -1)

Reorder the terms:
2 + 4x = 1 + (1 + x)(-1 + x)

Multiply (1 + x) * (-1 + x)
2 + 4x = 1 + (1(-1 + x) + x(-1 + x))
2 + 4x = 1 + ((-1 * 1 + x * 1) + x(-1 + x))
2 + 4x = 1 + ((-1 + 1x) + x(-1 + x))
2 + 4x = 1 + (-1 + 1x + (-1 * x + x * x))
2 + 4x = 1 + (-1 + 1x + (-1x + x2))

Combine like terms: 1x + -1x = 0
2 + 4x = 1 + (-1 + 0 + x2)
2 + 4x = 1 + (-1 + x2)

Combine like terms: 1 + -1 = 0
2 + 4x = 0 + x2
2 + 4x = x2

Solving
2 + 4x = x2

Solving for variable 'x'.

Combine like terms: x2 + -1x2 = 0
2 + 4x + -1x2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-2 + -4x + x2 = 0

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + -4x + 2 + x2 = 0 + 2

Reorder the terms:
-2 + 2 + -4x + x2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + -4x + x2 = 0 + 2
-4x + x2 = 0 + 2

Combine like terms: 0 + 2 = 2
-4x + x2 = 2

The x term is -4x.  Take half its coefficient (-2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
-4x + 4 + x2 = 2 + 4

Reorder the terms:
4 + -4x + x2 = 2 + 4

Combine like terms: 2 + 4 = 6
4 + -4x + x2 = 6

Factor a perfect square on the left side:
(x + -2)(x + -2) = 6

Calculate the square root of the right side: 2.449489743

Break this problem into two subproblems by setting 
(x + -2) equal to 2.449489743 and -2.449489743.


Subproblem 1
x + -2 = 2.449489743

Simplifying
x + -2 = 2.449489743

Reorder the terms:
-2 + x = 2.449489743

Solving
-2 + x = 2.449489743

Solving for variable 'x'.

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

Add '2' to each side of the equation.
-2 + 2 + x = 2.449489743 + 2

Combine like terms: -2 + 2 = 0
0 + x = 2.449489743 + 2
x = 2.449489743 + 2

Combine like terms: 2.449489743 + 2 = 4.449489743
x = 4.449489743

Simplifying
x = 4.449489743


Subproblem 2
x + -2 = -2.449489743

Simplifying
x + -2 = -2.449489743

Reorder the terms:
-2 + x = -2.449489743

Solving
-2 + x = -2.449489743

Solving for variable 'x'.

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

Add '2' to each side of the equation.
-2 + 2 + x = -2.449489743 + 2

Combine like terms: -2 + 2 = 0
0 + x = -2.449489743 + 2
x = -2.449489743 + 2

Combine like terms: -2.449489743 + 2 = -0.449489743
x = -0.449489743

Simplifying
x = -0.449489743


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {4.449489743, -0.449489743}

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