9n^2-6n-11=0

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Solution for 9n^2-6n-11=0 equation: 


Simplifying
9n2 + -6n + -11 = 0

Reorder the terms:
-11 + -6n + 9n2 = 0

Solving
-11 + -6n + 9n2 = 0

Solving for variable 'n'.

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

Divide each side by '9'.
-1.222222222 + -0.6666666667n + n2 = 0

Move the constant term to the right:

Add '1.222222222' to each side of the equation.
-1.222222222 + -0.6666666667n + 1.222222222 + n2 = 0 + 1.222222222

Reorder the terms:
-1.222222222 + 1.222222222 + -0.6666666667n + n2 = 0 + 1.222222222

Combine like terms: -1.222222222 + 1.222222222 = 0.000000000
0.000000000 + -0.6666666667n + n2 = 0 + 1.222222222
-0.6666666667n + n2 = 0 + 1.222222222

Combine like terms: 0 + 1.222222222 = 1.222222222
-0.6666666667n + n2 = 1.222222222

The n term is -0.6666666667n.  Take half its coefficient (-0.3333333334).
Square it (0.1111111112) and add it to both sides.

Add '0.1111111112' to each side of the equation.
-0.6666666667n + 0.1111111112 + n2 = 1.222222222 + 0.1111111112

Reorder the terms:
0.1111111112 + -0.6666666667n + n2 = 1.222222222 + 0.1111111112

Combine like terms: 1.222222222 + 0.1111111112 = 1.3333333332
0.1111111112 + -0.6666666667n + n2 = 1.3333333332

Factor a perfect square on the left side:
(n + -0.3333333334)(n + -0.3333333334) = 1.3333333332

Calculate the square root of the right side: 1.154700538

Break this problem into two subproblems by setting 
(n + -0.3333333334) equal to 1.154700538 and -1.154700538.


Subproblem 1
n + -0.3333333334 = 1.154700538

Simplifying
n + -0.3333333334 = 1.154700538

Reorder the terms:
-0.3333333334 + n = 1.154700538

Solving
-0.3333333334 + n = 1.154700538

Solving for variable 'n'.

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

Add '0.3333333334' to each side of the equation.
-0.3333333334 + 0.3333333334 + n = 1.154700538 + 0.3333333334

Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000
0.0000000000 + n = 1.154700538 + 0.3333333334
n = 1.154700538 + 0.3333333334

Combine like terms: 1.154700538 + 0.3333333334 = 1.4880338714
n = 1.4880338714

Simplifying
n = 1.4880338714


Subproblem 2
n + -0.3333333334 = -1.154700538

Simplifying
n + -0.3333333334 = -1.154700538

Reorder the terms:
-0.3333333334 + n = -1.154700538

Solving
-0.3333333334 + n = -1.154700538

Solving for variable 'n'.

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

Add '0.3333333334' to each side of the equation.
-0.3333333334 + 0.3333333334 + n = -1.154700538 + 0.3333333334

Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000
0.0000000000 + n = -1.154700538 + 0.3333333334
n = -1.154700538 + 0.3333333334

Combine like terms: -1.154700538 + 0.3333333334 = -0.8213672046
n = -0.8213672046

Simplifying
n = -0.8213672046


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {1.4880338714, -0.8213672046}

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