(2n+1)n=1257

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Solution for (2n+1)n=1257 equation: 


Simplifying
(2n + 1) * n = 1257

Reorder the terms:
(1 + 2n) * n = 1257

Reorder the terms for easier multiplication:
n(1 + 2n) = 1257
(1 * n + 2n * n) = 1257
(1n + 2n2) = 1257

Solving
1n + 2n2 = 1257

Solving for variable 'n'.

Reorder the terms:
-1257 + 1n + 2n2 = 1257 + -1257

Combine like terms: 1257 + -1257 = 0
-1257 + 1n + 2n2 = 0

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

Divide each side by '2'.
-628.5 + 0.5n + n2 = 0

Move the constant term to the right:

Add '628.5' to each side of the equation.
-628.5 + 0.5n + 628.5 + n2 = 0 + 628.5

Reorder the terms:
-628.5 + 628.5 + 0.5n + n2 = 0 + 628.5

Combine like terms: -628.5 + 628.5 = 0.0
0.0 + 0.5n + n2 = 0 + 628.5
0.5n + n2 = 0 + 628.5

Combine like terms: 0 + 628.5 = 628.5
0.5n + n2 = 628.5

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 = 628.5 + 0.0625

Reorder the terms:
0.0625 + 0.5n + n2 = 628.5 + 0.0625

Combine like terms: 628.5 + 0.0625 = 628.5625
0.0625 + 0.5n + n2 = 628.5625

Factor a perfect square on the left side:
(n + 0.25)(n + 0.25) = 628.5625

Calculate the square root of the right side: 25.071148757

Break this problem into two subproblems by setting 
(n + 0.25) equal to 25.071148757 and -25.071148757.


Subproblem 1
n + 0.25 = 25.071148757

Simplifying
n + 0.25 = 25.071148757

Reorder the terms:
0.25 + n = 25.071148757

Solving
0.25 + n = 25.071148757

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 = 25.071148757 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + n = 25.071148757 + -0.25
n = 25.071148757 + -0.25

Combine like terms: 25.071148757 + -0.25 = 24.821148757
n = 24.821148757

Simplifying
n = 24.821148757


Subproblem 2
n + 0.25 = -25.071148757

Simplifying
n + 0.25 = -25.071148757

Reorder the terms:
0.25 + n = -25.071148757

Solving
0.25 + n = -25.071148757

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 = -25.071148757 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + n = -25.071148757 + -0.25
n = -25.071148757 + -0.25

Combine like terms: -25.071148757 + -0.25 = -25.321148757
n = -25.321148757

Simplifying
n = -25.321148757


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {24.821148757, -25.321148757}

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