(2x+1)(2x-1)-(2x+1)2=8

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Solution for (2x+1)(2x-1)-(2x+1)2=8 equation:


Simplifying
(2x + 1)(2x + -1) + -1(2x + 1) * 2 = 8

Reorder the terms:
(1 + 2x)(2x + -1) + -1(2x + 1) * 2 = 8

Reorder the terms:
(1 + 2x)(-1 + 2x) + -1(2x + 1) * 2 = 8

Multiply (1 + 2x) * (-1 + 2x)
(1(-1 + 2x) + 2x * (-1 + 2x)) + -1(2x + 1) * 2 = 8
((-1 * 1 + 2x * 1) + 2x * (-1 + 2x)) + -1(2x + 1) * 2 = 8
((-1 + 2x) + 2x * (-1 + 2x)) + -1(2x + 1) * 2 = 8
(-1 + 2x + (-1 * 2x + 2x * 2x)) + -1(2x + 1) * 2 = 8
(-1 + 2x + (-2x + 4x2)) + -1(2x + 1) * 2 = 8

Combine like terms: 2x + -2x = 0
(-1 + 0 + 4x2) + -1(2x + 1) * 2 = 8
(-1 + 4x2) + -1(2x + 1) * 2 = 8

Reorder the terms:
-1 + 4x2 + -1(1 + 2x) * 2 = 8

Reorder the terms for easier multiplication:
-1 + 4x2 + -1 * 2(1 + 2x) = 8

Multiply -1 * 2
-1 + 4x2 + -2(1 + 2x) = 8
-1 + 4x2 + (1 * -2 + 2x * -2) = 8
-1 + 4x2 + (-2 + -4x) = 8

Reorder the terms:
-1 + -2 + -4x + 4x2 = 8

Combine like terms: -1 + -2 = -3
-3 + -4x + 4x2 = 8

Solving
-3 + -4x + 4x2 = 8

Solving for variable 'x'.

Reorder the terms:
-3 + -8 + -4x + 4x2 = 8 + -8

Combine like terms: -3 + -8 = -11
-11 + -4x + 4x2 = 8 + -8

Combine like terms: 8 + -8 = 0
-11 + -4x + 4x2 = 0

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

Divide each side by '4'.
-2.75 + -1x + x2 = 0

Move the constant term to the right:

Add '2.75' to each side of the equation.
-2.75 + -1x + 2.75 + x2 = 0 + 2.75

Reorder the terms:
-2.75 + 2.75 + -1x + x2 = 0 + 2.75

Combine like terms: -2.75 + 2.75 = 0.00
0.00 + -1x + x2 = 0 + 2.75
-1x + x2 = 0 + 2.75

Combine like terms: 0 + 2.75 = 2.75
-1x + x2 = 2.75

The x term is -1x.  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.
-1x + 0.25 + x2 = 2.75 + 0.25

Reorder the terms:
0.25 + -1x + x2 = 2.75 + 0.25

Combine like terms: 2.75 + 0.25 = 3
0.25 + -1x + x2 = 3

Factor a perfect square on the left side:
(x + -0.5)(x + -0.5) = 3

Calculate the square root of the right side: 1.732050808

Break this problem into two subproblems by setting 
(x + -0.5) equal to 1.732050808 and -1.732050808.

Subproblem 1

x + -0.5 = 1.732050808 Simplifying x + -0.5 = 1.732050808 Reorder the terms: -0.5 + x = 1.732050808 Solving -0.5 + x = 1.732050808 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 = 1.732050808 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = 1.732050808 + 0.5 x = 1.732050808 + 0.5 Combine like terms: 1.732050808 + 0.5 = 2.232050808 x = 2.232050808 Simplifying x = 2.232050808

Subproblem 2

x + -0.5 = -1.732050808 Simplifying x + -0.5 = -1.732050808 Reorder the terms: -0.5 + x = -1.732050808 Solving -0.5 + x = -1.732050808 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 = -1.732050808 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = -1.732050808 + 0.5 x = -1.732050808 + 0.5 Combine like terms: -1.732050808 + 0.5 = -1.232050808 x = -1.232050808 Simplifying x = -1.232050808

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.232050808, -1.232050808}

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