(2n+0.5)n=1

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Solution for (2n+0.5)n=1 equation: 


Simplifying
(2n + 0.5) * n = 1

Reorder the terms:
(0.5 + 2n) * n = 1

Reorder the terms for easier multiplication:
n(0.5 + 2n) = 1
(0.5 * n + 2n * n) = 1
(0.5n + 2n2) = 1

Solving
0.5n + 2n2 = 1

Solving for variable 'n'.

Reorder the terms:
-1 + 0.5n + 2n2 = 1 + -1

Combine like terms: 1 + -1 = 0
-1 + 0.5n + 2n2 = 0

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

Divide each side by '2'.
-0.5 + 0.25n + n2 = 0

Move the constant term to the right:

Add '0.5' to each side of the equation.
-0.5 + 0.25n + 0.5 + n2 = 0 + 0.5

Reorder the terms:
-0.5 + 0.5 + 0.25n + n2 = 0 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + 0.25n + n2 = 0 + 0.5
0.25n + n2 = 0 + 0.5

Combine like terms: 0 + 0.5 = 0.5
0.25n + n2 = 0.5

The n term is 0.25n.  Take half its coefficient (0.125).
Square it (0.015625) and add it to both sides.

Add '0.015625' to each side of the equation.
0.25n + 0.015625 + n2 = 0.5 + 0.015625

Reorder the terms:
0.015625 + 0.25n + n2 = 0.5 + 0.015625

Combine like terms: 0.5 + 0.015625 = 0.515625
0.015625 + 0.25n + n2 = 0.515625

Factor a perfect square on the left side:
(n + 0.125)(n + 0.125) = 0.515625

Calculate the square root of the right side: 0.718070331

Break this problem into two subproblems by setting 
(n + 0.125) equal to 0.718070331 and -0.718070331.


Subproblem 1
n + 0.125 = 0.718070331

Simplifying
n + 0.125 = 0.718070331

Reorder the terms:
0.125 + n = 0.718070331

Solving
0.125 + n = 0.718070331

Solving for variable 'n'.

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

Add '-0.125' to each side of the equation.
0.125 + -0.125 + n = 0.718070331 + -0.125

Combine like terms: 0.125 + -0.125 = 0.000
0.000 + n = 0.718070331 + -0.125
n = 0.718070331 + -0.125

Combine like terms: 0.718070331 + -0.125 = 0.593070331
n = 0.593070331

Simplifying
n = 0.593070331


Subproblem 2
n + 0.125 = -0.718070331

Simplifying
n + 0.125 = -0.718070331

Reorder the terms:
0.125 + n = -0.718070331

Solving
0.125 + n = -0.718070331

Solving for variable 'n'.

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

Add '-0.125' to each side of the equation.
0.125 + -0.125 + n = -0.718070331 + -0.125

Combine like terms: 0.125 + -0.125 = 0.000
0.000 + n = -0.718070331 + -0.125
n = -0.718070331 + -0.125

Combine like terms: -0.718070331 + -0.125 = -0.843070331
n = -0.843070331

Simplifying
n = -0.843070331


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {0.593070331, -0.843070331}

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