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

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

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

Reorder the terms:
-2n(-5 + n) = 3n + -1(n + 2)
(-5 * -2n + n * -2n) = 3n + -1(n + 2)
(10n + -2n2) = 3n + -1(n + 2)

Reorder the terms:
10n + -2n2 = 3n + -1(2 + n)
10n + -2n2 = 3n + (2 * -1 + n * -1)
10n + -2n2 = 3n + (-2 + -1n)

Reorder the terms:
10n + -2n2 = -2 + 3n + -1n

Combine like terms: 3n + -1n = 2n
10n + -2n2 = -2 + 2n

Solving
10n + -2n2 = -2 + 2n

Solving for variable 'n'.

Reorder the terms:
2 + 10n + -2n + -2n2 = -2 + 2n + 2 + -2n

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

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

Combine like terms: -2 + 2 = 0
2 + 8n + -2n2 = 0 + 2n + -2n
2 + 8n + -2n2 = 2n + -2n

Combine like terms: 2n + -2n = 0
2 + 8n + -2n2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(1 + 4n + -1n2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(1 + 4n + -1n2)' equal to zero and attempt to solve:

Simplifying
1 + 4n + -1n2 = 0

Solving
1 + 4n + -1n2 = 0

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

Divide each side by '-1'.
-1 + -4n + n2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -4n + 1 + n2 = 0 + 1

Reorder the terms:
-1 + 1 + -4n + n2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -4n + n2 = 0 + 1
-4n + n2 = 0 + 1

Combine like terms: 0 + 1 = 1
-4n + n2 = 1

The n term is -4n.  Take half its coefficient (-2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
-4n + 4 + n2 = 1 + 4

Reorder the terms:
4 + -4n + n2 = 1 + 4

Combine like terms: 1 + 4 = 5
4 + -4n + n2 = 5

Factor a perfect square on the left side:
(n + -2)(n + -2) = 5

Calculate the square root of the right side: 2.236067978

Break this problem into two subproblems by setting 
(n + -2) equal to 2.236067978 and -2.236067978.

Subproblem 1
n + -2 = 2.236067978

Simplifying
n + -2 = 2.236067978

Reorder the terms:
-2 + n = 2.236067978

Solving
-2 + n = 2.236067978

Solving for variable 'n'.

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

Add '2' to each side of the equation.
-2 + 2 + n = 2.236067978 + 2

Combine like terms: -2 + 2 = 0
0 + n = 2.236067978 + 2
n = 2.236067978 + 2

Combine like terms: 2.236067978 + 2 = 4.236067978
n = 4.236067978

Simplifying
n = 4.236067978

Subproblem 2
n + -2 = -2.236067978

Simplifying
n + -2 = -2.236067978

Reorder the terms:
-2 + n = -2.236067978

Solving
-2 + n = -2.236067978

Solving for variable 'n'.

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

Add '2' to each side of the equation.
-2 + 2 + n = -2.236067978 + 2

Combine like terms: -2 + 2 = 0
0 + n = -2.236067978 + 2
n = -2.236067978 + 2

Combine like terms: -2.236067978 + 2 = -0.236067978
n = -0.236067978

Simplifying
n = -0.236067978

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {4.236067978, -0.236067978}
Solution
n = {4.236067978, -0.236067978}