8(1+3n)+n(-4n+4)=

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Solution for 8(1+3n)+n(-4n+4)= equation: 


Simplifying
8(1 + 3n) + n(-4n + 4) = 0
(1 * 8 + 3n * 8) + n(-4n + 4) = 0
(8 + 24n) + n(-4n + 4) = 0

Reorder the terms:
8 + 24n + n(4 + -4n) = 0
8 + 24n + (4 * n + -4n * n) = 0
8 + 24n + (4n + -4n2) = 0

Combine like terms: 24n + 4n = 28n
8 + 28n + -4n2 = 0

Solving
8 + 28n + -4n2 = 0

Solving for variable 'n'.

Factor out the Greatest Common Factor (GCF), '4'.
4(2 + 7n + -1n2) = 0

Ignore the factor 4.

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

Simplifying
2 + 7n + -1n2 = 0

Solving
2 + 7n + -1n2 = 0

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

Divide each side by '-1'.
-2 + -7n + n2 = 0

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + -7n + 2 + n2 = 0 + 2

Reorder the terms:
-2 + 2 + -7n + n2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + -7n + n2 = 0 + 2
-7n + n2 = 0 + 2

Combine like terms: 0 + 2 = 2
-7n + n2 = 2

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 = 2 + 12.25

Reorder the terms:
12.25 + -7n + n2 = 2 + 12.25

Combine like terms: 2 + 12.25 = 14.25
12.25 + -7n + n2 = 14.25

Factor a perfect square on the left side:
(n + -3.5)(n + -3.5) = 14.25

Calculate the square root of the right side: 3.774917218

Break this problem into two subproblems by setting 
(n + -3.5) equal to 3.774917218 and -3.774917218.


Subproblem 1
n + -3.5 = 3.774917218

Simplifying
n + -3.5 = 3.774917218

Reorder the terms:
-3.5 + n = 3.774917218

Solving
-3.5 + n = 3.774917218

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 = 3.774917218 + 3.5

Combine like terms: -3.5 + 3.5 = 0.0
0.0 + n = 3.774917218 + 3.5
n = 3.774917218 + 3.5

Combine like terms: 3.774917218 + 3.5 = 7.274917218
n = 7.274917218

Simplifying
n = 7.274917218


Subproblem 2
n + -3.5 = -3.774917218

Simplifying
n + -3.5 = -3.774917218

Reorder the terms:
-3.5 + n = -3.774917218

Solving
-3.5 + n = -3.774917218

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 = -3.774917218 + 3.5

Combine like terms: -3.5 + 3.5 = 0.0
0.0 + n = -3.774917218 + 3.5
n = -3.774917218 + 3.5

Combine like terms: -3.774917218 + 3.5 = -0.274917218
n = -0.274917218

Simplifying
n = -0.274917218


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {7.274917218, -0.274917218}
Solution
n = {7.274917218, -0.274917218}

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