n(n+1)=370.5

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Solution for n(n+1)=370.5 equation: 


Simplifying
n(n + 1) = 370.5

Reorder the terms:
n(1 + n) = 370.5
(1 * n + n * n) = 370.5
(1n + n2) = 370.5

Solving
1n + n2 = 370.5

Solving for variable 'n'.

Reorder the terms:
-370.5 + 1n + n2 = 370.5 + -370.5

Combine like terms: 370.5 + -370.5 = 0.0
-370.5 + 1n + n2 = 0.0

Begin completing the square.

Move the constant term to the right:

Add '370.5' to each side of the equation.
-370.5 + 1n + 370.5 + n2 = 0.0 + 370.5

Reorder the terms:
-370.5 + 370.5 + 1n + n2 = 0.0 + 370.5

Combine like terms: -370.5 + 370.5 = 0.0
0.0 + 1n + n2 = 0.0 + 370.5
1n + n2 = 0.0 + 370.5

Combine like terms: 0.0 + 370.5 = 370.5
1n + n2 = 370.5

The n term is 1n.  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.
n + 0.25 + n2 = 370.5 + 0.25

Reorder the terms:
0.25 + n + n2 = 370.5 + 0.25

Combine like terms: 370.5 + 0.25 = 370.75
0.25 + n + n2 = 370.75

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 370.75

Calculate the square root of the right side: 19.254869514

Break this problem into two subproblems by setting 
(n + 0.5) equal to 19.254869514 and -19.254869514.


Subproblem 1
n + 0.5 = 19.254869514

Simplifying
n + 0.5 = 19.254869514

Reorder the terms:
0.5 + n = 19.254869514

Solving
0.5 + n = 19.254869514

Solving for variable 'n'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + n = 19.254869514 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + n = 19.254869514 + -0.5
n = 19.254869514 + -0.5

Combine like terms: 19.254869514 + -0.5 = 18.754869514
n = 18.754869514

Simplifying
n = 18.754869514


Subproblem 2
n + 0.5 = -19.254869514

Simplifying
n + 0.5 = -19.254869514

Reorder the terms:
0.5 + n = -19.254869514

Solving
0.5 + n = -19.254869514

Solving for variable 'n'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + n = -19.254869514 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + n = -19.254869514 + -0.5
n = -19.254869514 + -0.5

Combine like terms: -19.254869514 + -0.5 = -19.754869514
n = -19.754869514

Simplifying
n = -19.754869514


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {18.754869514, -19.754869514}

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