12n^2-11n-2=0

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Solution for 12n^2-11n-2=0 equation: 


Simplifying
12n2 + -11n + -2 = 0

Reorder the terms:
-2 + -11n + 12n2 = 0

Solving
-2 + -11n + 12n2 = 0

Solving for variable 'n'.

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

Divide each side by '12'.
-0.1666666667 + -0.9166666667n + n2 = 0

Move the constant term to the right:

Add '0.1666666667' to each side of the equation.
-0.1666666667 + -0.9166666667n + 0.1666666667 + n2 = 0 + 0.1666666667

Reorder the terms:
-0.1666666667 + 0.1666666667 + -0.9166666667n + n2 = 0 + 0.1666666667

Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000
0.0000000000 + -0.9166666667n + n2 = 0 + 0.1666666667
-0.9166666667n + n2 = 0 + 0.1666666667

Combine like terms: 0 + 0.1666666667 = 0.1666666667
-0.9166666667n + n2 = 0.1666666667

The n term is -0.9166666667n.  Take half its coefficient (-0.4583333334).
Square it (0.2100694445) and add it to both sides.

Add '0.2100694445' to each side of the equation.
-0.9166666667n + 0.2100694445 + n2 = 0.1666666667 + 0.2100694445

Reorder the terms:
0.2100694445 + -0.9166666667n + n2 = 0.1666666667 + 0.2100694445

Combine like terms: 0.1666666667 + 0.2100694445 = 0.3767361112
0.2100694445 + -0.9166666667n + n2 = 0.3767361112

Factor a perfect square on the left side:
(n + -0.4583333334)(n + -0.4583333334) = 0.3767361112

Calculate the square root of the right side: 0.613788328

Break this problem into two subproblems by setting 
(n + -0.4583333334) equal to 0.613788328 and -0.613788328.


Subproblem 1
n + -0.4583333334 = 0.613788328

Simplifying
n + -0.4583333334 = 0.613788328

Reorder the terms:
-0.4583333334 + n = 0.613788328

Solving
-0.4583333334 + n = 0.613788328

Solving for variable 'n'.

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

Add '0.4583333334' to each side of the equation.
-0.4583333334 + 0.4583333334 + n = 0.613788328 + 0.4583333334

Combine like terms: -0.4583333334 + 0.4583333334 = 0.0000000000
0.0000000000 + n = 0.613788328 + 0.4583333334
n = 0.613788328 + 0.4583333334

Combine like terms: 0.613788328 + 0.4583333334 = 1.0721216614
n = 1.0721216614

Simplifying
n = 1.0721216614


Subproblem 2
n + -0.4583333334 = -0.613788328

Simplifying
n + -0.4583333334 = -0.613788328

Reorder the terms:
-0.4583333334 + n = -0.613788328

Solving
-0.4583333334 + n = -0.613788328

Solving for variable 'n'.

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

Add '0.4583333334' to each side of the equation.
-0.4583333334 + 0.4583333334 + n = -0.613788328 + 0.4583333334

Combine like terms: -0.4583333334 + 0.4583333334 = 0.0000000000
0.0000000000 + n = -0.613788328 + 0.4583333334
n = -0.613788328 + 0.4583333334

Combine like terms: -0.613788328 + 0.4583333334 = -0.1554549946
n = -0.1554549946

Simplifying
n = -0.1554549946


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {1.0721216614, -0.1554549946}

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