12n^2-10n-9=0

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Solution for 12n^2-10n-9=0 equation: 


Simplifying
12n2 + -10n + -9 = 0

Reorder the terms:
-9 + -10n + 12n2 = 0

Solving
-9 + -10n + 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'.
-0.75 + -0.8333333333n + n2 = 0

Move the constant term to the right:

Add '0.75' to each side of the equation.
-0.75 + -0.8333333333n + 0.75 + n2 = 0 + 0.75

Reorder the terms:
-0.75 + 0.75 + -0.8333333333n + n2 = 0 + 0.75

Combine like terms: -0.75 + 0.75 = 0.00
0.00 + -0.8333333333n + n2 = 0 + 0.75
-0.8333333333n + n2 = 0 + 0.75

Combine like terms: 0 + 0.75 = 0.75
-0.8333333333n + n2 = 0.75

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 = 0.75 + 0.1736111111

Reorder the terms:
0.1736111111 + -0.8333333333n + n2 = 0.75 + 0.1736111111

Combine like terms: 0.75 + 0.1736111111 = 0.9236111111
0.1736111111 + -0.8333333333n + n2 = 0.9236111111

Factor a perfect square on the left side:
(n + -0.4166666667)(n + -0.4166666667) = 0.9236111111

Calculate the square root of the right side: 0.961046883

Break this problem into two subproblems by setting 
(n + -0.4166666667) equal to 0.961046883 and -0.961046883.


Subproblem 1
n + -0.4166666667 = 0.961046883

Simplifying
n + -0.4166666667 = 0.961046883

Reorder the terms:
-0.4166666667 + n = 0.961046883

Solving
-0.4166666667 + n = 0.961046883

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 = 0.961046883 + 0.4166666667

Combine like terms: -0.4166666667 + 0.4166666667 = 0.0000000000
0.0000000000 + n = 0.961046883 + 0.4166666667
n = 0.961046883 + 0.4166666667

Combine like terms: 0.961046883 + 0.4166666667 = 1.3777135497
n = 1.3777135497

Simplifying
n = 1.3777135497


Subproblem 2
n + -0.4166666667 = -0.961046883

Simplifying
n + -0.4166666667 = -0.961046883

Reorder the terms:
-0.4166666667 + n = -0.961046883

Solving
-0.4166666667 + n = -0.961046883

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 = -0.961046883 + 0.4166666667

Combine like terms: -0.4166666667 + 0.4166666667 = 0.0000000000
0.0000000000 + n = -0.961046883 + 0.4166666667
n = -0.961046883 + 0.4166666667

Combine like terms: -0.961046883 + 0.4166666667 = -0.5443802163
n = -0.5443802163

Simplifying
n = -0.5443802163


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {1.3777135497, -0.5443802163}

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