(n+1)(n+1)=133

Solution for (n+1)(n+1)=133 equation:

Simplifying
(n + 1)(n + 1) = 133

Reorder the terms:
(1 + n)(n + 1) = 133

Reorder the terms:
(1 + n)(1 + n) = 133

Multiply (1 + n) * (1 + n)
(1(1 + n) + n(1 + n)) = 133
((1 * 1 + n * 1) + n(1 + n)) = 133
((1 + 1n) + n(1 + n)) = 133
(1 + 1n + (1 * n + n * n)) = 133
(1 + 1n + (1n + n2)) = 133

Combine like terms: 1n + 1n = 2n
(1 + 2n + n2) = 133

Solving
1 + 2n + n2 = 133

Solving for variable 'n'.

Reorder the terms:
1 + -133 + 2n + n2 = 133 + -133

Combine like terms: 1 + -133 = -132
-132 + 2n + n2 = 133 + -133

Combine like terms: 133 + -133 = 0
-132 + 2n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '132' to each side of the equation.
-132 + 2n + 132 + n2 = 0 + 132

Reorder the terms:
-132 + 132 + 2n + n2 = 0 + 132

Combine like terms: -132 + 132 = 0
0 + 2n + n2 = 0 + 132
2n + n2 = 0 + 132

Combine like terms: 0 + 132 = 132
2n + n2 = 132

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 = 132 + 1

Reorder the terms:
1 + 2n + n2 = 132 + 1

Combine like terms: 132 + 1 = 133
1 + 2n + n2 = 133

Factor a perfect square on the left side:
(n + 1)(n + 1) = 133

Calculate the square root of the right side: 11.532562595

Break this problem into two subproblems by setting 
(n + 1) equal to 11.532562595 and -11.532562595.

Subproblem 1
n + 1 = 11.532562595

Simplifying
n + 1 = 11.532562595

Reorder the terms:
1 + n = 11.532562595

Solving
1 + n = 11.532562595

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 = 11.532562595 + -1

Combine like terms: 1 + -1 = 0
0 + n = 11.532562595 + -1
n = 11.532562595 + -1

Combine like terms: 11.532562595 + -1 = 10.532562595
n = 10.532562595

Simplifying
n = 10.532562595

Subproblem 2
n + 1 = -11.532562595

Simplifying
n + 1 = -11.532562595

Reorder the terms:
1 + n = -11.532562595

Solving
1 + n = -11.532562595

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 = -11.532562595 + -1

Combine like terms: 1 + -1 = 0
0 + n = -11.532562595 + -1
n = -11.532562595 + -1

Combine like terms: -11.532562595 + -1 = -12.532562595
n = -12.532562595

Simplifying
n = -12.532562595

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {10.532562595, -12.532562595}