6n^2-18n-16=6

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Solution for 6n^2-18n-16=6 equation: 


Simplifying
6n2 + -18n + -16 = 6

Reorder the terms:
-16 + -18n + 6n2 = 6

Solving
-16 + -18n + 6n2 = 6

Solving for variable 'n'.

Reorder the terms:
-16 + -6 + -18n + 6n2 = 6 + -6

Combine like terms: -16 + -6 = -22
-22 + -18n + 6n2 = 6 + -6

Combine like terms: 6 + -6 = 0
-22 + -18n + 6n2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-11 + -9n + 3n2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-11 + -9n + 3n2)' equal to zero and attempt to solve:

Simplifying
-11 + -9n + 3n2 = 0

Solving
-11 + -9n + 3n2 = 0

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

Divide each side by '3'.
-3.666666667 + -3n + n2 = 0

Move the constant term to the right:

Add '3.666666667' to each side of the equation.
-3.666666667 + -3n + 3.666666667 + n2 = 0 + 3.666666667

Reorder the terms:
-3.666666667 + 3.666666667 + -3n + n2 = 0 + 3.666666667

Combine like terms: -3.666666667 + 3.666666667 = 0.000000000
0.000000000 + -3n + n2 = 0 + 3.666666667
-3n + n2 = 0 + 3.666666667

Combine like terms: 0 + 3.666666667 = 3.666666667
-3n + n2 = 3.666666667

The n term is -3n.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3n + 2.25 + n2 = 3.666666667 + 2.25

Reorder the terms:
2.25 + -3n + n2 = 3.666666667 + 2.25

Combine like terms: 3.666666667 + 2.25 = 5.916666667
2.25 + -3n + n2 = 5.916666667

Factor a perfect square on the left side:
(n + -1.5)(n + -1.5) = 5.916666667

Calculate the square root of the right side: 2.43241992

Break this problem into two subproblems by setting 
(n + -1.5) equal to 2.43241992 and -2.43241992.


Subproblem 1
n + -1.5 = 2.43241992

Simplifying
n + -1.5 = 2.43241992

Reorder the terms:
-1.5 + n = 2.43241992

Solving
-1.5 + n = 2.43241992

Solving for variable 'n'.

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

Add '1.5' to each side of the equation.
-1.5 + 1.5 + n = 2.43241992 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + n = 2.43241992 + 1.5
n = 2.43241992 + 1.5

Combine like terms: 2.43241992 + 1.5 = 3.93241992
n = 3.93241992

Simplifying
n = 3.93241992


Subproblem 2
n + -1.5 = -2.43241992

Simplifying
n + -1.5 = -2.43241992

Reorder the terms:
-1.5 + n = -2.43241992

Solving
-1.5 + n = -2.43241992

Solving for variable 'n'.

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

Add '1.5' to each side of the equation.
-1.5 + 1.5 + n = -2.43241992 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + n = -2.43241992 + 1.5
n = -2.43241992 + 1.5

Combine like terms: -2.43241992 + 1.5 = -0.93241992
n = -0.93241992

Simplifying
n = -0.93241992


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {3.93241992, -0.93241992}
Solution
n = {3.93241992, -0.93241992}

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