2n+1=n^2-1

Solution for 2n+1=n^2-1 equation:

Simplifying
2n + 1 = n2 + -1

Reorder the terms:
1 + 2n = n2 + -1

Reorder the terms:
1 + 2n = -1 + n2

Solving
1 + 2n = -1 + n2

Solving for variable 'n'.

Reorder the terms:
1 + 1 + 2n + -1n2 = -1 + n2 + 1 + -1n2

Combine like terms: 1 + 1 = 2
2 + 2n + -1n2 = -1 + n2 + 1 + -1n2

Reorder the terms:
2 + 2n + -1n2 = -1 + 1 + n2 + -1n2

Combine like terms: -1 + 1 = 0
2 + 2n + -1n2 = 0 + n2 + -1n2
2 + 2n + -1n2 = n2 + -1n2

Combine like terms: n2 + -1n2 = 0
2 + 2n + -1n2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-2 + -2n + n2 = 0

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + -2n + 2 + n2 = 0 + 2

Reorder the terms:
-2 + 2 + -2n + n2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + -2n + n2 = 0 + 2
-2n + n2 = 0 + 2

Combine like terms: 0 + 2 = 2
-2n + n2 = 2

The n term is -2n.  Take half its coefficient (-1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
-2n + 1 + n2 = 2 + 1

Reorder the terms:
1 + -2n + n2 = 2 + 1

Combine like terms: 2 + 1 = 3
1 + -2n + n2 = 3

Factor a perfect square on the left side:
(n + -1)(n + -1) = 3

Calculate the square root of the right side: 1.732050808

Break this problem into two subproblems by setting 
(n + -1) equal to 1.732050808 and -1.732050808.

Subproblem 1
n + -1 = 1.732050808

Simplifying
n + -1 = 1.732050808

Reorder the terms:
-1 + n = 1.732050808

Solving
-1 + n = 1.732050808

Solving for variable 'n'.

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

Add '1' to each side of the equation.
-1 + 1 + n = 1.732050808 + 1

Combine like terms: -1 + 1 = 0
0 + n = 1.732050808 + 1
n = 1.732050808 + 1

Combine like terms: 1.732050808 + 1 = 2.732050808
n = 2.732050808

Simplifying
n = 2.732050808

Subproblem 2
n + -1 = -1.732050808

Simplifying
n + -1 = -1.732050808

Reorder the terms:
-1 + n = -1.732050808

Solving
-1 + n = -1.732050808

Solving for variable 'n'.

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

Add '1' to each side of the equation.
-1 + 1 + n = -1.732050808 + 1

Combine like terms: -1 + 1 = 0
0 + n = -1.732050808 + 1
n = -1.732050808 + 1

Combine like terms: -1.732050808 + 1 = -0.732050808
n = -0.732050808

Simplifying
n = -0.732050808

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {2.732050808, -0.732050808}