n(n-2)=83+4(n+n+2)

Solution for n(n-2)=83+4(n+n+2) equation:

Simplifying
n(n + -2) = 83 + 4(n + n + 2)

Reorder the terms:
n(-2 + n) = 83 + 4(n + n + 2)
(-2 * n + n * n) = 83 + 4(n + n + 2)
(-2n + n2) = 83 + 4(n + n + 2)

Reorder the terms:
-2n + n2 = 83 + 4(2 + n + n)

Combine like terms: n + n = 2n
-2n + n2 = 83 + 4(2 + 2n)
-2n + n2 = 83 + (2 * 4 + 2n * 4)
-2n + n2 = 83 + (8 + 8n)

Combine like terms: 83 + 8 = 91
-2n + n2 = 91 + 8n

Solving
-2n + n2 = 91 + 8n

Solving for variable 'n'.

Reorder the terms:
-91 + -2n + -8n + n2 = 91 + 8n + -91 + -8n

Combine like terms: -2n + -8n = -10n
-91 + -10n + n2 = 91 + 8n + -91 + -8n

Reorder the terms:
-91 + -10n + n2 = 91 + -91 + 8n + -8n

Combine like terms: 91 + -91 = 0
-91 + -10n + n2 = 0 + 8n + -8n
-91 + -10n + n2 = 8n + -8n

Combine like terms: 8n + -8n = 0
-91 + -10n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '91' to each side of the equation.
-91 + -10n + 91 + n2 = 0 + 91

Reorder the terms:
-91 + 91 + -10n + n2 = 0 + 91

Combine like terms: -91 + 91 = 0
0 + -10n + n2 = 0 + 91
-10n + n2 = 0 + 91

Combine like terms: 0 + 91 = 91
-10n + n2 = 91

The n term is -10n.  Take half its coefficient (-5).
Square it (25) and add it to both sides.

Add '25' to each side of the equation.
-10n + 25 + n2 = 91 + 25

Reorder the terms:
25 + -10n + n2 = 91 + 25

Combine like terms: 91 + 25 = 116
25 + -10n + n2 = 116

Factor a perfect square on the left side:
(n + -5)(n + -5) = 116

Calculate the square root of the right side: 10.770329614

Break this problem into two subproblems by setting 
(n + -5) equal to 10.770329614 and -10.770329614.

Subproblem 1
n + -5 = 10.770329614

Simplifying
n + -5 = 10.770329614

Reorder the terms:
-5 + n = 10.770329614

Solving
-5 + n = 10.770329614

Solving for variable 'n'.

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

Add '5' to each side of the equation.
-5 + 5 + n = 10.770329614 + 5

Combine like terms: -5 + 5 = 0
0 + n = 10.770329614 + 5
n = 10.770329614 + 5

Combine like terms: 10.770329614 + 5 = 15.770329614
n = 15.770329614

Simplifying
n = 15.770329614

Subproblem 2
n + -5 = -10.770329614

Simplifying
n + -5 = -10.770329614

Reorder the terms:
-5 + n = -10.770329614

Solving
-5 + n = -10.770329614

Solving for variable 'n'.

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

Add '5' to each side of the equation.
-5 + 5 + n = -10.770329614 + 5

Combine like terms: -5 + 5 = 0
0 + n = -10.770329614 + 5
n = -10.770329614 + 5

Combine like terms: -10.770329614 + 5 = -5.770329614
n = -5.770329614

Simplifying
n = -5.770329614

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {15.770329614, -5.770329614}