9n^2-9n-21=0

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Solution for 9n^2-9n-21=0 equation: 


Simplifying
9n2 + -9n + -21 = 0

Reorder the terms:
-21 + -9n + 9n2 = 0

Solving
-21 + -9n + 9n2 = 0

Solving for variable 'n'.

Factor out the Greatest Common Factor (GCF), '3'.
3(-7 + -3n + 3n2) = 0

Ignore the factor 3.

Subproblem 1
Set the factor '(-7 + -3n + 3n2)' equal to zero and attempt to solve:

Simplifying
-7 + -3n + 3n2 = 0

Solving
-7 + -3n + 3n2 = 0

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

Divide each side by '3'.
-2.333333333 + -1n + n2 = 0

Move the constant term to the right:

Add '2.333333333' to each side of the equation.
-2.333333333 + -1n + 2.333333333 + n2 = 0 + 2.333333333

Reorder the terms:
-2.333333333 + 2.333333333 + -1n + n2 = 0 + 2.333333333

Combine like terms: -2.333333333 + 2.333333333 = 0.000000000
0.000000000 + -1n + n2 = 0 + 2.333333333
-1n + n2 = 0 + 2.333333333

Combine like terms: 0 + 2.333333333 = 2.333333333
-1n + n2 = 2.333333333

The n term is -1n.  Take half its coefficient (-0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1n + 0.25 + n2 = 2.333333333 + 0.25

Reorder the terms:
0.25 + -1n + n2 = 2.333333333 + 0.25

Combine like terms: 2.333333333 + 0.25 = 2.583333333
0.25 + -1n + n2 = 2.583333333

Factor a perfect square on the left side:
(n + -0.5)(n + -0.5) = 2.583333333

Calculate the square root of the right side: 1.607275127

Break this problem into two subproblems by setting 
(n + -0.5) equal to 1.607275127 and -1.607275127.


Subproblem 1
n + -0.5 = 1.607275127

Simplifying
n + -0.5 = 1.607275127

Reorder the terms:
-0.5 + n = 1.607275127

Solving
-0.5 + n = 1.607275127

Solving for variable 'n'.

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

Add '0.5' to each side of the equation.
-0.5 + 0.5 + n = 1.607275127 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + n = 1.607275127 + 0.5
n = 1.607275127 + 0.5

Combine like terms: 1.607275127 + 0.5 = 2.107275127
n = 2.107275127

Simplifying
n = 2.107275127


Subproblem 2
n + -0.5 = -1.607275127

Simplifying
n + -0.5 = -1.607275127

Reorder the terms:
-0.5 + n = -1.607275127

Solving
-0.5 + n = -1.607275127

Solving for variable 'n'.

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

Add '0.5' to each side of the equation.
-0.5 + 0.5 + n = -1.607275127 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + n = -1.607275127 + 0.5
n = -1.607275127 + 0.5

Combine like terms: -1.607275127 + 0.5 = -1.107275127
n = -1.107275127

Simplifying
n = -1.107275127


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {2.107275127, -1.107275127}
Solution
n = {2.107275127, -1.107275127}

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