n(n-1)=32

Solution for n(n-1)=32 equation:

Simplifying
n(n + -1) = 32

Reorder the terms:
n(-1 + n) = 32
(-1 * n + n * n) = 32
(-1n + n2) = 32

Solving
-1n + n2 = 32

Solving for variable 'n'.

Reorder the terms:
-32 + -1n + n2 = 32 + -32

Combine like terms: 32 + -32 = 0
-32 + -1n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '32' to each side of the equation.
-32 + -1n + 32 + n2 = 0 + 32

Reorder the terms:
-32 + 32 + -1n + n2 = 0 + 32

Combine like terms: -32 + 32 = 0
0 + -1n + n2 = 0 + 32
-1n + n2 = 0 + 32

Combine like terms: 0 + 32 = 32
-1n + n2 = 32

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 = 32 + 0.25

Reorder the terms:
0.25 + -1n + n2 = 32 + 0.25

Combine like terms: 32 + 0.25 = 32.25
0.25 + -1n + n2 = 32.25

Factor a perfect square on the left side:
(n + -0.5)(n + -0.5) = 32.25

Calculate the square root of the right side: 5.678908346

Break this problem into two subproblems by setting 
(n + -0.5) equal to 5.678908346 and -5.678908346.

Subproblem 1
n + -0.5 = 5.678908346

Simplifying
n + -0.5 = 5.678908346

Reorder the terms:
-0.5 + n = 5.678908346

Solving
-0.5 + n = 5.678908346

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 = 5.678908346 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + n = 5.678908346 + 0.5
n = 5.678908346 + 0.5

Combine like terms: 5.678908346 + 0.5 = 6.178908346
n = 6.178908346

Simplifying
n = 6.178908346

Subproblem 2
n + -0.5 = -5.678908346

Simplifying
n + -0.5 = -5.678908346

Reorder the terms:
-0.5 + n = -5.678908346

Solving
-0.5 + n = -5.678908346

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 = -5.678908346 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + n = -5.678908346 + 0.5
n = -5.678908346 + 0.5

Combine like terms: -5.678908346 + 0.5 = -5.178908346
n = -5.178908346

Simplifying
n = -5.178908346

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {6.178908346, -5.178908346}