8n(6n+5)=7

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Solution for 8n(6n+5)=7 equation: 


Simplifying
8n(6n + 5) = 7

Reorder the terms:
8n(5 + 6n) = 7
(5 * 8n + 6n * 8n) = 7
(40n + 48n2) = 7

Solving
40n + 48n2 = 7

Solving for variable 'n'.

Reorder the terms:
-7 + 40n + 48n2 = 7 + -7

Combine like terms: 7 + -7 = 0
-7 + 40n + 48n2 = 0

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

Divide each side by '48'.
-0.1458333333 + 0.8333333333n + n2 = 0

Move the constant term to the right:

Add '0.1458333333' to each side of the equation.
-0.1458333333 + 0.8333333333n + 0.1458333333 + n2 = 0 + 0.1458333333

Reorder the terms:
-0.1458333333 + 0.1458333333 + 0.8333333333n + n2 = 0 + 0.1458333333

Combine like terms: -0.1458333333 + 0.1458333333 = 0.0000000000
0.0000000000 + 0.8333333333n + n2 = 0 + 0.1458333333
0.8333333333n + n2 = 0 + 0.1458333333

Combine like terms: 0 + 0.1458333333 = 0.1458333333
0.8333333333n + n2 = 0.1458333333

The n term is 0.8333333333n.  Take half its coefficient (0.4166666667).
Square it (0.1736111111) and add it to both sides.

Add '0.1736111111' to each side of the equation.
0.8333333333n + 0.1736111111 + n2 = 0.1458333333 + 0.1736111111

Reorder the terms:
0.1736111111 + 0.8333333333n + n2 = 0.1458333333 + 0.1736111111

Combine like terms: 0.1458333333 + 0.1736111111 = 0.3194444444
0.1736111111 + 0.8333333333n + n2 = 0.3194444444

Factor a perfect square on the left side:
(n + 0.4166666667)(n + 0.4166666667) = 0.3194444444

Calculate the square root of the right side: 0.565194165

Break this problem into two subproblems by setting 
(n + 0.4166666667) equal to 0.565194165 and -0.565194165.


Subproblem 1
n + 0.4166666667 = 0.565194165

Simplifying
n + 0.4166666667 = 0.565194165

Reorder the terms:
0.4166666667 + n = 0.565194165

Solving
0.4166666667 + n = 0.565194165

Solving for variable 'n'.

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

Add '-0.4166666667' to each side of the equation.
0.4166666667 + -0.4166666667 + n = 0.565194165 + -0.4166666667

Combine like terms: 0.4166666667 + -0.4166666667 = 0.0000000000
0.0000000000 + n = 0.565194165 + -0.4166666667
n = 0.565194165 + -0.4166666667

Combine like terms: 0.565194165 + -0.4166666667 = 0.1485274983
n = 0.1485274983

Simplifying
n = 0.1485274983


Subproblem 2
n + 0.4166666667 = -0.565194165

Simplifying
n + 0.4166666667 = -0.565194165

Reorder the terms:
0.4166666667 + n = -0.565194165

Solving
0.4166666667 + n = -0.565194165

Solving for variable 'n'.

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

Add '-0.4166666667' to each side of the equation.
0.4166666667 + -0.4166666667 + n = -0.565194165 + -0.4166666667

Combine like terms: 0.4166666667 + -0.4166666667 = 0.0000000000
0.0000000000 + n = -0.565194165 + -0.4166666667
n = -0.565194165 + -0.4166666667

Combine like terms: -0.565194165 + -0.4166666667 = -0.9818608317
n = -0.9818608317

Simplifying
n = -0.9818608317


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {0.1485274983, -0.9818608317}

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