2n(n-7)+6n=6(n+2)

Solution for 2n(n-7)+6n=6(n+2) equation:

Simplifying
2n(n + -7) + 6n = 6(n + 2)

Reorder the terms:
2n(-7 + n) + 6n = 6(n + 2)
(-7 * 2n + n * 2n) + 6n = 6(n + 2)
(-14n + 2n2) + 6n = 6(n + 2)

Reorder the terms:
-14n + 6n + 2n2 = 6(n + 2)

Combine like terms: -14n + 6n = -8n
-8n + 2n2 = 6(n + 2)

Reorder the terms:
-8n + 2n2 = 6(2 + n)
-8n + 2n2 = (2 * 6 + n * 6)
-8n + 2n2 = (12 + 6n)

Solving
-8n + 2n2 = 12 + 6n

Solving for variable 'n'.

Reorder the terms:
-12 + -8n + -6n + 2n2 = 12 + 6n + -12 + -6n

Combine like terms: -8n + -6n = -14n
-12 + -14n + 2n2 = 12 + 6n + -12 + -6n

Reorder the terms:
-12 + -14n + 2n2 = 12 + -12 + 6n + -6n

Combine like terms: 12 + -12 = 0
-12 + -14n + 2n2 = 0 + 6n + -6n
-12 + -14n + 2n2 = 6n + -6n

Combine like terms: 6n + -6n = 0
-12 + -14n + 2n2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-6 + -7n + n2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-6 + -7n + n2)' equal to zero and attempt to solve:

Simplifying
-6 + -7n + n2 = 0

Solving
-6 + -7n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '6' to each side of the equation.
-6 + -7n + 6 + n2 = 0 + 6

Reorder the terms:
-6 + 6 + -7n + n2 = 0 + 6

Combine like terms: -6 + 6 = 0
0 + -7n + n2 = 0 + 6
-7n + n2 = 0 + 6

Combine like terms: 0 + 6 = 6
-7n + n2 = 6

The n term is -7n.  Take half its coefficient (-3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
-7n + 12.25 + n2 = 6 + 12.25

Reorder the terms:
12.25 + -7n + n2 = 6 + 12.25

Combine like terms: 6 + 12.25 = 18.25
12.25 + -7n + n2 = 18.25

Factor a perfect square on the left side:
(n + -3.5)(n + -3.5) = 18.25

Calculate the square root of the right side: 4.272001873

Break this problem into two subproblems by setting 
(n + -3.5) equal to 4.272001873 and -4.272001873.

Subproblem 1
n + -3.5 = 4.272001873

Simplifying
n + -3.5 = 4.272001873

Reorder the terms:
-3.5 + n = 4.272001873

Solving
-3.5 + n = 4.272001873

Solving for variable 'n'.

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

Add '3.5' to each side of the equation.
-3.5 + 3.5 + n = 4.272001873 + 3.5

Combine like terms: -3.5 + 3.5 = 0.0
0.0 + n = 4.272001873 + 3.5
n = 4.272001873 + 3.5

Combine like terms: 4.272001873 + 3.5 = 7.772001873
n = 7.772001873

Simplifying
n = 7.772001873

Subproblem 2
n + -3.5 = -4.272001873

Simplifying
n + -3.5 = -4.272001873

Reorder the terms:
-3.5 + n = -4.272001873

Solving
-3.5 + n = -4.272001873

Solving for variable 'n'.

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

Add '3.5' to each side of the equation.
-3.5 + 3.5 + n = -4.272001873 + 3.5

Combine like terms: -3.5 + 3.5 = 0.0
0.0 + n = -4.272001873 + 3.5
n = -4.272001873 + 3.5

Combine like terms: -4.272001873 + 3.5 = -0.772001873
n = -0.772001873

Simplifying
n = -0.772001873

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {7.772001873, -0.772001873}
Solution
n = {7.772001873, -0.772001873}