(2x+1)(2x)=64

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Solution for (2x+1)(2x)=64 equation: 


Simplifying
(2x + 1)(2x) = 64

Reorder the terms:
(1 + 2x)(2x) = 64

Remove parenthesis around (2x)
(1 + 2x) * 2x = 64

Reorder the terms for easier multiplication:
2x(1 + 2x) = 64
(1 * 2x + 2x * 2x) = 64
(2x + 4x2) = 64

Solving
2x + 4x2 = 64

Solving for variable 'x'.

Reorder the terms:
-64 + 2x + 4x2 = 64 + -64

Combine like terms: 64 + -64 = 0
-64 + 2x + 4x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-32 + x + 2x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-32 + x + 2x2)' equal to zero and attempt to solve:

Simplifying
-32 + x + 2x2 = 0

Solving
-32 + x + 2x2 = 0

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

Divide each side by '2'.
-16 + 0.5x + x2 = 0

Move the constant term to the right:

Add '16' to each side of the equation.
-16 + 0.5x + 16 + x2 = 0 + 16

Reorder the terms:
-16 + 16 + 0.5x + x2 = 0 + 16

Combine like terms: -16 + 16 = 0
0 + 0.5x + x2 = 0 + 16
0.5x + x2 = 0 + 16

Combine like terms: 0 + 16 = 16
0.5x + x2 = 16

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.
0.5x + 0.25 + x2 = 16 + 0.25

Reorder the terms:
0.25 + 0.5x + x2 = 16 + 0.25

Combine like terms: 16 + 0.25 = 16.25
0.25 + 0.5x + x2 = 16.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 16.25

Calculate the square root of the right side: 4.031128874

Break this problem into two subproblems by setting 
(x + 0.5) equal to 4.031128874 and -4.031128874.


Subproblem 1
x + 0.5 = 4.031128874

Simplifying
x + 0.5 = 4.031128874

Reorder the terms:
0.5 + x = 4.031128874

Solving
0.5 + x = 4.031128874

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 = 4.031128874 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 4.031128874 + -0.5
x = 4.031128874 + -0.5

Combine like terms: 4.031128874 + -0.5 = 3.531128874
x = 3.531128874

Simplifying
x = 3.531128874


Subproblem 2
x + 0.5 = -4.031128874

Simplifying
x + 0.5 = -4.031128874

Reorder the terms:
0.5 + x = -4.031128874

Solving
0.5 + x = -4.031128874

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 = -4.031128874 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -4.031128874 + -0.5
x = -4.031128874 + -0.5

Combine like terms: -4.031128874 + -0.5 = -4.531128874
x = -4.531128874

Simplifying
x = -4.531128874


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.531128874, -4.531128874}
Solution
x = {3.531128874, -4.531128874}

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