-6n(5+8n)=-366

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Solution for -6n(5+8n)=-366 equation:


Simplifying
-6n(5 + 8n) = -366
(5 * -6n + 8n * -6n) = -366
(-30n + -48n2) = -366

Solving
-30n + -48n2 = -366

Solving for variable 'n'.

Reorder the terms:
366 + -30n + -48n2 = -366 + 366

Combine like terms: -366 + 366 = 0
366 + -30n + -48n2 = 0

Factor out the Greatest Common Factor (GCF), '6'.
6(61 + -5n + -8n2) = 0

Ignore the factor 6.

Subproblem 1
Set the factor '(61 + -5n + -8n2)' equal to zero and attempt to solve:

Simplifying
61 + -5n + -8n2 = 0

Solving
61 + -5n + -8n2 = 0

Begin completing the square.  Divide all terms by
-8 the coefficient of the squared term: 

Divide each side by '-8'.
-7.625 + 0.625n + n2 = 0

Move the constant term to the right:

Add '7.625' to each side of the equation.
-7.625 + 0.625n + 7.625 + n2 = 0 + 7.625

Reorder the terms:
-7.625 + 7.625 + 0.625n + n2 = 0 + 7.625

Combine like terms: -7.625 + 7.625 = 0.000
0.000 + 0.625n + n2 = 0 + 7.625
0.625n + n2 = 0 + 7.625

Combine like terms: 0 + 7.625 = 7.625
0.625n + n2 = 7.625

The n term is 0.625n.  Take half its coefficient (0.3125).
Square it (0.09765625) and add it to both sides.

Add '0.09765625' to each side of the equation.
0.625n + 0.09765625 + n2 = 7.625 + 0.09765625

Reorder the terms:
0.09765625 + 0.625n + n2 = 7.625 + 0.09765625

Combine like terms: 7.625 + 0.09765625 = 7.72265625
0.09765625 + 0.625n + n2 = 7.72265625

Factor a perfect square on the left side:
(n + 0.3125)(n + 0.3125) = 7.72265625

Calculate the square root of the right side: 2.778966759

Break this problem into two subproblems by setting 
(n + 0.3125) equal to 2.778966759 and -2.778966759.


Subproblem 1
n + 0.3125 = 2.778966759

Simplifying
n + 0.3125 = 2.778966759

Reorder the terms:
0.3125 + n = 2.778966759

Solving
0.3125 + n = 2.778966759

Solving for variable 'n'.

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

Add '-0.3125' to each side of the equation.
0.3125 + -0.3125 + n = 2.778966759 + -0.3125

Combine like terms: 0.3125 + -0.3125 = 0.0000
0.0000 + n = 2.778966759 + -0.3125
n = 2.778966759 + -0.3125

Combine like terms: 2.778966759 + -0.3125 = 2.466466759
n = 2.466466759

Simplifying
n = 2.466466759


Subproblem 2
n + 0.3125 = -2.778966759

Simplifying
n + 0.3125 = -2.778966759

Reorder the terms:
0.3125 + n = -2.778966759

Solving
0.3125 + n = -2.778966759

Solving for variable 'n'.

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

Add '-0.3125' to each side of the equation.
0.3125 + -0.3125 + n = -2.778966759 + -0.3125

Combine like terms: 0.3125 + -0.3125 = 0.0000
0.0000 + n = -2.778966759 + -0.3125
n = -2.778966759 + -0.3125

Combine like terms: -2.778966759 + -0.3125 = -3.091466759
n = -3.091466759

Simplifying
n = -3.091466759


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {2.466466759, -3.091466759}
Solution
n = {2.466466759, -3.091466759}

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