2n(n-1)-n=3n+16

Solution for 2n(n-1)-n=3n+16 equation:

Simplifying
2n(n + -1) + -1n = 3n + 16

Reorder the terms:
2n(-1 + n) + -1n = 3n + 16
(-1 * 2n + n * 2n) + -1n = 3n + 16
(-2n + 2n2) + -1n = 3n + 16

Reorder the terms:
-2n + -1n + 2n2 = 3n + 16

Combine like terms: -2n + -1n = -3n
-3n + 2n2 = 3n + 16

Reorder the terms:
-3n + 2n2 = 16 + 3n

Solving
-3n + 2n2 = 16 + 3n

Solving for variable 'n'.

Reorder the terms:
-16 + -3n + -3n + 2n2 = 16 + 3n + -16 + -3n

Combine like terms: -3n + -3n = -6n
-16 + -6n + 2n2 = 16 + 3n + -16 + -3n

Reorder the terms:
-16 + -6n + 2n2 = 16 + -16 + 3n + -3n

Combine like terms: 16 + -16 = 0
-16 + -6n + 2n2 = 0 + 3n + -3n
-16 + -6n + 2n2 = 3n + -3n

Combine like terms: 3n + -3n = 0
-16 + -6n + 2n2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-8 + -3n + n2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-8 + -3n + n2)' equal to zero and attempt to solve:

Simplifying
-8 + -3n + n2 = 0

Solving
-8 + -3n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '8' to each side of the equation.
-8 + -3n + 8 + n2 = 0 + 8

Reorder the terms:
-8 + 8 + -3n + n2 = 0 + 8

Combine like terms: -8 + 8 = 0
0 + -3n + n2 = 0 + 8
-3n + n2 = 0 + 8

Combine like terms: 0 + 8 = 8
-3n + n2 = 8

The n term is -3n.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3n + 2.25 + n2 = 8 + 2.25

Reorder the terms:
2.25 + -3n + n2 = 8 + 2.25

Combine like terms: 8 + 2.25 = 10.25
2.25 + -3n + n2 = 10.25

Factor a perfect square on the left side:
(n + -1.5)(n + -1.5) = 10.25

Calculate the square root of the right side: 3.201562119

Break this problem into two subproblems by setting 
(n + -1.5) equal to 3.201562119 and -3.201562119.

Subproblem 1
n + -1.5 = 3.201562119

Simplifying
n + -1.5 = 3.201562119

Reorder the terms:
-1.5 + n = 3.201562119

Solving
-1.5 + n = 3.201562119

Solving for variable 'n'.

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

Add '1.5' to each side of the equation.
-1.5 + 1.5 + n = 3.201562119 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + n = 3.201562119 + 1.5
n = 3.201562119 + 1.5

Combine like terms: 3.201562119 + 1.5 = 4.701562119
n = 4.701562119

Simplifying
n = 4.701562119

Subproblem 2
n + -1.5 = -3.201562119

Simplifying
n + -1.5 = -3.201562119

Reorder the terms:
-1.5 + n = -3.201562119

Solving
-1.5 + n = -3.201562119

Solving for variable 'n'.

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

Add '1.5' to each side of the equation.
-1.5 + 1.5 + n = -3.201562119 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + n = -3.201562119 + 1.5
n = -3.201562119 + 1.5

Combine like terms: -3.201562119 + 1.5 = -1.701562119
n = -1.701562119

Simplifying
n = -1.701562119

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {4.701562119, -1.701562119}
Solution
n = {4.701562119, -1.701562119}