n=4+7n(54+n)

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Solution for n=4+7n(54+n) equation:


Simplifying
n = 4 + 7n(54 + n)
n = 4 + (54 * 7n + n * 7n)
n = 4 + (378n + 7n2)

Solving
n = 4 + 378n + 7n2

Solving for variable 'n'.

Reorder the terms:
-4 + n + -378n + -7n2 = 4 + 378n + 7n2 + -4 + -378n + -7n2

Combine like terms: n + -378n = -377n
-4 + -377n + -7n2 = 4 + 378n + 7n2 + -4 + -378n + -7n2

Reorder the terms:
-4 + -377n + -7n2 = 4 + -4 + 378n + -378n + 7n2 + -7n2

Combine like terms: 4 + -4 = 0
-4 + -377n + -7n2 = 0 + 378n + -378n + 7n2 + -7n2
-4 + -377n + -7n2 = 378n + -378n + 7n2 + -7n2

Combine like terms: 378n + -378n = 0
-4 + -377n + -7n2 = 0 + 7n2 + -7n2
-4 + -377n + -7n2 = 7n2 + -7n2

Combine like terms: 7n2 + -7n2 = 0
-4 + -377n + -7n2 = 0

Factor out the Greatest Common Factor (GCF), '-1'.
-1(4 + 377n + 7n2) = 0

Ignore the factor -1.

Subproblem 1
Set the factor '(4 + 377n + 7n2)' equal to zero and attempt to solve:

Simplifying
4 + 377n + 7n2 = 0

Solving
4 + 377n + 7n2 = 0

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

Divide each side by '7'.
0.5714285714 + 53.85714286n + n2 = 0

Move the constant term to the right:

Add '-0.5714285714' to each side of the equation.
0.5714285714 + 53.85714286n + -0.5714285714 + n2 = 0 + -0.5714285714

Reorder the terms:
0.5714285714 + -0.5714285714 + 53.85714286n + n2 = 0 + -0.5714285714

Combine like terms: 0.5714285714 + -0.5714285714 = 0.0000000000
0.0000000000 + 53.85714286n + n2 = 0 + -0.5714285714
53.85714286n + n2 = 0 + -0.5714285714

Combine like terms: 0 + -0.5714285714 = -0.5714285714
53.85714286n + n2 = -0.5714285714

The n term is 53.85714286n.  Take half its coefficient (26.92857143).
Square it (725.1479593) and add it to both sides.

Add '725.1479593' to each side of the equation.
53.85714286n + 725.1479593 + n2 = -0.5714285714 + 725.1479593

Reorder the terms:
725.1479593 + 53.85714286n + n2 = -0.5714285714 + 725.1479593

Combine like terms: -0.5714285714 + 725.1479593 = 724.5765307286
725.1479593 + 53.85714286n + n2 = 724.5765307286

Factor a perfect square on the left side:
(n + 26.92857143)(n + 26.92857143) = 724.5765307286

Calculate the square root of the right side: 26.91795926

Break this problem into two subproblems by setting 
(n + 26.92857143) equal to 26.91795926 and -26.91795926.


Subproblem 1
n + 26.92857143 = 26.91795926

Simplifying
n + 26.92857143 = 26.91795926

Reorder the terms:
26.92857143 + n = 26.91795926

Solving
26.92857143 + n = 26.91795926

Solving for variable 'n'.

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

Add '-26.92857143' to each side of the equation.
26.92857143 + -26.92857143 + n = 26.91795926 + -26.92857143

Combine like terms: 26.92857143 + -26.92857143 = 0.00000000
0.00000000 + n = 26.91795926 + -26.92857143
n = 26.91795926 + -26.92857143

Combine like terms: 26.91795926 + -26.92857143 = -0.01061217
n = -0.01061217

Simplifying
n = -0.01061217


Subproblem 2
n + 26.92857143 = -26.91795926

Simplifying
n + 26.92857143 = -26.91795926

Reorder the terms:
26.92857143 + n = -26.91795926

Solving
26.92857143 + n = -26.91795926

Solving for variable 'n'.

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

Add '-26.92857143' to each side of the equation.
26.92857143 + -26.92857143 + n = -26.91795926 + -26.92857143

Combine like terms: 26.92857143 + -26.92857143 = 0.00000000
0.00000000 + n = -26.91795926 + -26.92857143
n = -26.91795926 + -26.92857143

Combine like terms: -26.91795926 + -26.92857143 = -53.84653069
n = -53.84653069

Simplifying
n = -53.84653069


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {-0.01061217, -53.84653069}
Solution
n = {-0.01061217, -53.84653069}

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