0.5(n+1)(n-2)=35

Solution for 0.5(n+1)(n-2)=35 equation:

Simplifying
0.5(n + 1)(n + -2) = 35

Reorder the terms:
0.5(1 + n)(n + -2) = 35

Reorder the terms:
0.5(1 + n)(-2 + n) = 35

Multiply (1 + n) * (-2 + n)
0.5(1(-2 + n) + n(-2 + n)) = 35
0.5((-2 * 1 + n * 1) + n(-2 + n)) = 35
0.5((-2 + 1n) + n(-2 + n)) = 35
0.5(-2 + 1n + (-2 * n + n * n)) = 35
0.5(-2 + 1n + (-2n + n2)) = 35

Combine like terms: 1n + -2n = -1n
0.5(-2 + -1n + n2) = 35
(-2 * 0.5 + -1n * 0.5 + n2 * 0.5) = 35
(-1 + -0.5n + 0.5n2) = 35

Solving
-1 + -0.5n + 0.5n2 = 35

Solving for variable 'n'.

Reorder the terms:
-1 + -35 + -0.5n + 0.5n2 = 35 + -35

Combine like terms: -1 + -35 = -36
-36 + -0.5n + 0.5n2 = 35 + -35

Combine like terms: 35 + -35 = 0
-36 + -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'.
-72 + -1n + n2 = 0

Move the constant term to the right:

Add '72' to each side of the equation.
-72 + -1n + 72 + n2 = 0 + 72

Reorder the terms:
-72 + 72 + -1n + n2 = 0 + 72

Combine like terms: -72 + 72 = 0
0 + -1n + n2 = 0 + 72
-1n + n2 = 0 + 72

Combine like terms: 0 + 72 = 72
-1n + n2 = 72

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.
-1n + 0.25 + n2 = 72 + 0.25

Reorder the terms:
0.25 + -1n + n2 = 72 + 0.25

Combine like terms: 72 + 0.25 = 72.25
0.25 + -1n + n2 = 72.25

Factor a perfect square on the left side:
(n + -0.5)(n + -0.5) = 72.25

Calculate the square root of the right side: 8.5

Break this problem into two subproblems by setting 
(n + -0.5) equal to 8.5 and -8.5.

Subproblem 1
n + -0.5 = 8.5

Simplifying
n + -0.5 = 8.5

Reorder the terms:
-0.5 + n = 8.5

Solving
-0.5 + n = 8.5

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 = 8.5 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + n = 8.5 + 0.5
n = 8.5 + 0.5

Combine like terms: 8.5 + 0.5 = 9
n = 9

Simplifying
n = 9

Subproblem 2
n + -0.5 = -8.5

Simplifying
n + -0.5 = -8.5

Reorder the terms:
-0.5 + n = -8.5

Solving
-0.5 + n = -8.5

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 = -8.5 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + n = -8.5 + 0.5
n = -8.5 + 0.5

Combine like terms: -8.5 + 0.5 = -8
n = -8

Simplifying
n = -8

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {9, -8}