8n^2+92n-364=0

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Solution for 8n^2+92n-364=0 equation: 


Simplifying
8n2 + 92n + -364 = 0

Reorder the terms:
-364 + 92n + 8n2 = 0

Solving
-364 + 92n + 8n2 = 0

Solving for variable 'n'.

Factor out the Greatest Common Factor (GCF), '4'.
4(-91 + 23n + 2n2) = 0

Ignore the factor 4.

Subproblem 1
Set the factor '(-91 + 23n + 2n2)' equal to zero and attempt to solve:

Simplifying
-91 + 23n + 2n2 = 0

Solving
-91 + 23n + 2n2 = 0

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

Divide each side by '2'.
-45.5 + 11.5n + n2 = 0

Move the constant term to the right:

Add '45.5' to each side of the equation.
-45.5 + 11.5n + 45.5 + n2 = 0 + 45.5

Reorder the terms:
-45.5 + 45.5 + 11.5n + n2 = 0 + 45.5

Combine like terms: -45.5 + 45.5 = 0.0
0.0 + 11.5n + n2 = 0 + 45.5
11.5n + n2 = 0 + 45.5

Combine like terms: 0 + 45.5 = 45.5
11.5n + n2 = 45.5

The n term is 11.5n.  Take half its coefficient (5.75).
Square it (33.0625) and add it to both sides.

Add '33.0625' to each side of the equation.
11.5n + 33.0625 + n2 = 45.5 + 33.0625

Reorder the terms:
33.0625 + 11.5n + n2 = 45.5 + 33.0625

Combine like terms: 45.5 + 33.0625 = 78.5625
33.0625 + 11.5n + n2 = 78.5625

Factor a perfect square on the left side:
(n + 5.75)(n + 5.75) = 78.5625

Calculate the square root of the right side: 8.863548951

Break this problem into two subproblems by setting 
(n + 5.75) equal to 8.863548951 and -8.863548951.


Subproblem 1
n + 5.75 = 8.863548951

Simplifying
n + 5.75 = 8.863548951

Reorder the terms:
5.75 + n = 8.863548951

Solving
5.75 + n = 8.863548951

Solving for variable 'n'.

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

Add '-5.75' to each side of the equation.
5.75 + -5.75 + n = 8.863548951 + -5.75

Combine like terms: 5.75 + -5.75 = 0.00
0.00 + n = 8.863548951 + -5.75
n = 8.863548951 + -5.75

Combine like terms: 8.863548951 + -5.75 = 3.113548951
n = 3.113548951

Simplifying
n = 3.113548951


Subproblem 2
n + 5.75 = -8.863548951

Simplifying
n + 5.75 = -8.863548951

Reorder the terms:
5.75 + n = -8.863548951

Solving
5.75 + n = -8.863548951

Solving for variable 'n'.

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

Add '-5.75' to each side of the equation.
5.75 + -5.75 + n = -8.863548951 + -5.75

Combine like terms: 5.75 + -5.75 = 0.00
0.00 + n = -8.863548951 + -5.75
n = -8.863548951 + -5.75

Combine like terms: -8.863548951 + -5.75 = -14.613548951
n = -14.613548951

Simplifying
n = -14.613548951


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {3.113548951, -14.613548951}
Solution
n = {3.113548951, -14.613548951}

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