0.5n(n+1)=8

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Solution for 0.5n(n+1)=8 equation: 


Simplifying
0.5n(n + 1) = 8

Reorder the terms:
0.5n(1 + n) = 8
(1 * 0.5n + n * 0.5n) = 8
(0.5n + 0.5n2) = 8

Solving
0.5n + 0.5n2 = 8

Solving for variable 'n'.

Reorder the terms:
-8 + 0.5n + 0.5n2 = 8 + -8

Combine like terms: 8 + -8 = 0
-8 + 0.5n + 0.5n2 = 0

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

Divide each side by '0.5'.
-16 + n + n2 = 0

Move the constant term to the right:

Add '16' to each side of the equation.
-16 + n + 16 + n2 = 0 + 16

Reorder the terms:
-16 + 16 + n + n2 = 0 + 16

Combine like terms: -16 + 16 = 0
0 + n + n2 = 0 + 16
n + n2 = 0 + 16

Combine like terms: 0 + 16 = 16
n + n2 = 16

The n term is n.  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.
 + 0.25 + n2 = 16 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + n2 = 16 + 0.25

Combine like terms: 16 + 0.25 = 16.25
1.25 + n2 = 16.25

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

Calculate the square root of the right side: 4.031128874

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


Subproblem 1
n + 0.5 = 4.031128874

Simplifying
n + 0.5 = 4.031128874

Reorder the terms:
0.5 + n = 4.031128874

Solving
0.5 + n = 4.031128874

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 = 4.031128874 + -0.5

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

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

Simplifying
n = 3.531128874


Subproblem 2
n + 0.5 = -4.031128874

Simplifying
n + 0.5 = -4.031128874

Reorder the terms:
0.5 + n = -4.031128874

Solving
0.5 + n = -4.031128874

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 = -4.031128874 + -0.5

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

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

Simplifying
n = -4.531128874


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

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