3n(n-1)=5n+3-2n

Solution for 3n(n-1)=5n+3-2n equation:

Simplifying
3n(n + -1) = 5n + 3 + -2n

Reorder the terms:
3n(-1 + n) = 5n + 3 + -2n
(-1 * 3n + n * 3n) = 5n + 3 + -2n
(-3n + 3n2) = 5n + 3 + -2n

Reorder the terms:
-3n + 3n2 = 3 + 5n + -2n

Combine like terms: 5n + -2n = 3n
-3n + 3n2 = 3 + 3n

Solving
-3n + 3n2 = 3 + 3n

Solving for variable 'n'.

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

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

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

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

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

Factor out the Greatest Common Factor (GCF), '3'.
3(-1 + -2n + n2) = 0

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

Simplifying
-1 + -2n + n2 = 0

Solving
-1 + -2n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -2n + 1 + n2 = 0 + 1

Reorder the terms:
-1 + 1 + -2n + n2 = 0 + 1

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

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

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 = 1 + 1

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

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

Factor a perfect square on the left side:
(n + -1)(n + -1) = 2

Calculate the square root of the right side: 1.414213562

Break this problem into two subproblems by setting 
(n + -1) equal to 1.414213562 and -1.414213562.

Subproblem 1
n + -1 = 1.414213562

Simplifying
n + -1 = 1.414213562

Reorder the terms:
-1 + n = 1.414213562

Solving
-1 + n = 1.414213562

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 = 1.414213562 + 1

Combine like terms: -1 + 1 = 0
0 + n = 1.414213562 + 1
n = 1.414213562 + 1

Combine like terms: 1.414213562 + 1 = 2.414213562
n = 2.414213562

Simplifying
n = 2.414213562

Subproblem 2
n + -1 = -1.414213562

Simplifying
n + -1 = -1.414213562

Reorder the terms:
-1 + n = -1.414213562

Solving
-1 + n = -1.414213562

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 = -1.414213562 + 1

Combine like terms: -1 + 1 = 0
0 + n = -1.414213562 + 1
n = -1.414213562 + 1

Combine like terms: -1.414213562 + 1 = -0.414213562
n = -0.414213562

Simplifying
n = -0.414213562

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {2.414213562, -0.414213562}
Solution
n = {2.414213562, -0.414213562}