0.5((n+1)(n-6))=30

Solution for 0.5((n+1)(n-6))=30 equation:

Simplifying
0.5((n + 1)(n + -6)) = 30

Reorder the terms:
0.5((1 + n)(n + -6)) = 30

Reorder the terms:
0.5((1 + n)(-6 + n)) = 30

Multiply (1 + n) * (-6 + n)
0.5((1(-6 + n) + n(-6 + n))) = 30
0.5(((-6 * 1 + n * 1) + n(-6 + n))) = 30
0.5(((-6 + 1n) + n(-6 + n))) = 30
0.5((-6 + 1n + (-6 * n + n * n))) = 30
0.5((-6 + 1n + (-6n + n2))) = 30

Combine like terms: 1n + -6n = -5n
0.5((-6 + -5n + n2)) = 30
(-6 * 0.5 + -5n * 0.5 + n2 * 0.5) = 30
(-3 + -2.5n + 0.5n2) = 30

Solving
-3 + -2.5n + 0.5n2 = 30

Solving for variable 'n'.

Reorder the terms:
-3 + -30 + -2.5n + 0.5n2 = 30 + -30

Combine like terms: -3 + -30 = -33
-33 + -2.5n + 0.5n2 = 30 + -30

Combine like terms: 30 + -30 = 0
-33 + -2.5n + 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'.
-66 + -5n + n2 = 0

Move the constant term to the right:

Add '66' to each side of the equation.
-66 + -5n + 66 + n2 = 0 + 66

Reorder the terms:
-66 + 66 + -5n + n2 = 0 + 66

Combine like terms: -66 + 66 = 0
0 + -5n + n2 = 0 + 66
-5n + n2 = 0 + 66

Combine like terms: 0 + 66 = 66
-5n + n2 = 66

The n term is -5n.  Take half its coefficient (-2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
-5n + 6.25 + n2 = 66 + 6.25

Reorder the terms:
6.25 + -5n + n2 = 66 + 6.25

Combine like terms: 66 + 6.25 = 72.25
6.25 + -5n + n2 = 72.25

Factor a perfect square on the left side:
(n + -2.5)(n + -2.5) = 72.25

Calculate the square root of the right side: 8.5

Break this problem into two subproblems by setting 
(n + -2.5) equal to 8.5 and -8.5.

Subproblem 1
n + -2.5 = 8.5

Simplifying
n + -2.5 = 8.5

Reorder the terms:
-2.5 + n = 8.5

Solving
-2.5 + n = 8.5

Solving for variable 'n'.

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

Add '2.5' to each side of the equation.
-2.5 + 2.5 + n = 8.5 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + n = 8.5 + 2.5
n = 8.5 + 2.5

Combine like terms: 8.5 + 2.5 = 11
n = 11

Simplifying
n = 11

Subproblem 2
n + -2.5 = -8.5

Simplifying
n + -2.5 = -8.5

Reorder the terms:
-2.5 + n = -8.5

Solving
-2.5 + n = -8.5

Solving for variable 'n'.

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

Add '2.5' to each side of the equation.
-2.5 + 2.5 + n = -8.5 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + n = -8.5 + 2.5
n = -8.5 + 2.5

Combine like terms: -8.5 + 2.5 = -6
n = -6

Simplifying
n = -6

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {11, -6}