(2n+1)n=355

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Solution for (2n+1)n=355 equation:


Simplifying
(2n + 1) * n = 355

Reorder the terms:
(1 + 2n) * n = 355

Reorder the terms for easier multiplication:
n(1 + 2n) = 355
(1 * n + 2n * n) = 355
(1n + 2n2) = 355

Solving
1n + 2n2 = 355

Solving for variable 'n'.

Reorder the terms:
-355 + 1n + 2n2 = 355 + -355

Combine like terms: 355 + -355 = 0
-355 + 1n + 2n2 = 0

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

Divide each side by '2'.
-177.5 + 0.5n + n2 = 0

Move the constant term to the right:

Add '177.5' to each side of the equation.
-177.5 + 0.5n + 177.5 + n2 = 0 + 177.5

Reorder the terms:
-177.5 + 177.5 + 0.5n + n2 = 0 + 177.5

Combine like terms: -177.5 + 177.5 = 0.0
0.0 + 0.5n + n2 = 0 + 177.5
0.5n + n2 = 0 + 177.5

Combine like terms: 0 + 177.5 = 177.5
0.5n + n2 = 177.5

The n term is 0.5n.  Take half its coefficient (0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
0.5n + 0.0625 + n2 = 177.5 + 0.0625

Reorder the terms:
0.0625 + 0.5n + n2 = 177.5 + 0.0625

Combine like terms: 177.5 + 0.0625 = 177.5625
0.0625 + 0.5n + n2 = 177.5625

Factor a perfect square on the left side:
(n + 0.25)(n + 0.25) = 177.5625

Calculate the square root of the right side: 13.325257971

Break this problem into two subproblems by setting 
(n + 0.25) equal to 13.325257971 and -13.325257971.

Subproblem 1

n + 0.25 = 13.325257971 Simplifying n + 0.25 = 13.325257971 Reorder the terms: 0.25 + n = 13.325257971 Solving 0.25 + n = 13.325257971 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + n = 13.325257971 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + n = 13.325257971 + -0.25 n = 13.325257971 + -0.25 Combine like terms: 13.325257971 + -0.25 = 13.075257971 n = 13.075257971 Simplifying n = 13.075257971

Subproblem 2

n + 0.25 = -13.325257971 Simplifying n + 0.25 = -13.325257971 Reorder the terms: 0.25 + n = -13.325257971 Solving 0.25 + n = -13.325257971 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + n = -13.325257971 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + n = -13.325257971 + -0.25 n = -13.325257971 + -0.25 Combine like terms: -13.325257971 + -0.25 = -13.575257971 n = -13.575257971 Simplifying n = -13.575257971

Solution

The solution to the problem is based on the solutions from the subproblems. n = {13.075257971, -13.575257971}

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