8n(5n+4)=14

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Solution for 8n(5n+4)=14 equation: 


Simplifying
8n(5n + 4) = 14

Reorder the terms:
8n(4 + 5n) = 14
(4 * 8n + 5n * 8n) = 14
(32n + 40n2) = 14

Solving
32n + 40n2 = 14

Solving for variable 'n'.

Reorder the terms:
-14 + 32n + 40n2 = 14 + -14

Combine like terms: 14 + -14 = 0
-14 + 32n + 40n2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-7 + 16n + 20n2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-7 + 16n + 20n2)' equal to zero and attempt to solve:

Simplifying
-7 + 16n + 20n2 = 0

Solving
-7 + 16n + 20n2 = 0

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

Divide each side by '20'.
-0.35 + 0.8n + n2 = 0.0

Move the constant term to the right:

Add '0.35' to each side of the equation.
-0.35 + 0.8n + 0.35 + n2 = 0.0 + 0.35

Reorder the terms:
-0.35 + 0.35 + 0.8n + n2 = 0.0 + 0.35

Combine like terms: -0.35 + 0.35 = 0.00
0.00 + 0.8n + n2 = 0.0 + 0.35
0.8n + n2 = 0.0 + 0.35

Combine like terms: 0.0 + 0.35 = 0.35
0.8n + n2 = 0.35

The n term is 0.8n.  Take half its coefficient (0.4).
Square it (0.16) and add it to both sides.

Add '0.16' to each side of the equation.
0.8n + 0.16 + n2 = 0.35 + 0.16

Reorder the terms:
0.16 + 0.8n + n2 = 0.35 + 0.16

Combine like terms: 0.35 + 0.16 = 0.51
0.16 + 0.8n + n2 = 0.51

Factor a perfect square on the left side:
(n + 0.4)(n + 0.4) = 0.51

Calculate the square root of the right side: 0.714142843

Break this problem into two subproblems by setting 
(n + 0.4) equal to 0.714142843 and -0.714142843.


Subproblem 1
n + 0.4 = 0.714142843

Simplifying
n + 0.4 = 0.714142843

Reorder the terms:
0.4 + n = 0.714142843

Solving
0.4 + n = 0.714142843

Solving for variable 'n'.

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

Add '-0.4' to each side of the equation.
0.4 + -0.4 + n = 0.714142843 + -0.4

Combine like terms: 0.4 + -0.4 = 0.0
0.0 + n = 0.714142843 + -0.4
n = 0.714142843 + -0.4

Combine like terms: 0.714142843 + -0.4 = 0.314142843
n = 0.314142843

Simplifying
n = 0.314142843


Subproblem 2
n + 0.4 = -0.714142843

Simplifying
n + 0.4 = -0.714142843

Reorder the terms:
0.4 + n = -0.714142843

Solving
0.4 + n = -0.714142843

Solving for variable 'n'.

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

Add '-0.4' to each side of the equation.
0.4 + -0.4 + n = -0.714142843 + -0.4

Combine like terms: 0.4 + -0.4 = 0.0
0.0 + n = -0.714142843 + -0.4
n = -0.714142843 + -0.4

Combine like terms: -0.714142843 + -0.4 = -1.114142843
n = -1.114142843

Simplifying
n = -1.114142843


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {0.314142843, -1.114142843}
Solution
n = {0.314142843, -1.114142843}

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