n(n-1)=1

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


Simplifying
n(n + -1) = 1

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

Solving
-1n + n2 = 1

Solving for variable 'n'.

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

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

Begin completing the square.

Move the constant term to the right:

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

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

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

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

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 = 1 + 0.25

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

Combine like terms: 1 + 0.25 = 1.25
0.25 + -1n + n2 = 1.25

Factor a perfect square on the left side:
(n + -0.5)(n + -0.5) = 1.25

Calculate the square root of the right side: 1.118033989

Break this problem into two subproblems by setting 
(n + -0.5) equal to 1.118033989 and -1.118033989.


Subproblem 1
n + -0.5 = 1.118033989

Simplifying
n + -0.5 = 1.118033989

Reorder the terms:
-0.5 + n = 1.118033989

Solving
-0.5 + n = 1.118033989

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 = 1.118033989 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + n = 1.118033989 + 0.5
n = 1.118033989 + 0.5

Combine like terms: 1.118033989 + 0.5 = 1.618033989
n = 1.618033989

Simplifying
n = 1.618033989


Subproblem 2
n + -0.5 = -1.118033989

Simplifying
n + -0.5 = -1.118033989

Reorder the terms:
-0.5 + n = -1.118033989

Solving
-0.5 + n = -1.118033989

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 = -1.118033989 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + n = -1.118033989 + 0.5
n = -1.118033989 + 0.5

Combine like terms: -1.118033989 + 0.5 = -0.618033989
n = -0.618033989

Simplifying
n = -0.618033989


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {1.618033989, -0.618033989}

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