9n^2-21n-162=0

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Solution for 9n^2-21n-162=0 equation: 


Simplifying
9n2 + -21n + -162 = 0

Reorder the terms:
-162 + -21n + 9n2 = 0

Solving
-162 + -21n + 9n2 = 0

Solving for variable 'n'.

Factor out the Greatest Common Factor (GCF), '3'.
3(-54 + -7n + 3n2) = 0

Ignore the factor 3.

Subproblem 1
Set the factor '(-54 + -7n + 3n2)' equal to zero and attempt to solve:

Simplifying
-54 + -7n + 3n2 = 0

Solving
-54 + -7n + 3n2 = 0

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

Divide each side by '3'.
-18 + -2.333333333n + n2 = 0

Move the constant term to the right:

Add '18' to each side of the equation.
-18 + -2.333333333n + 18 + n2 = 0 + 18

Reorder the terms:
-18 + 18 + -2.333333333n + n2 = 0 + 18

Combine like terms: -18 + 18 = 0
0 + -2.333333333n + n2 = 0 + 18
-2.333333333n + n2 = 0 + 18

Combine like terms: 0 + 18 = 18
-2.333333333n + n2 = 18

The n term is -2.333333333n.  Take half its coefficient (-1.166666667).
Square it (1.361111112) and add it to both sides.

Add '1.361111112' to each side of the equation.
-2.333333333n + 1.361111112 + n2 = 18 + 1.361111112

Reorder the terms:
1.361111112 + -2.333333333n + n2 = 18 + 1.361111112

Combine like terms: 18 + 1.361111112 = 19.361111112
1.361111112 + -2.333333333n + n2 = 19.361111112

Factor a perfect square on the left side:
(n + -1.166666667)(n + -1.166666667) = 19.361111112

Calculate the square root of the right side: 4.400126261

Break this problem into two subproblems by setting 
(n + -1.166666667) equal to 4.400126261 and -4.400126261.


Subproblem 1
n + -1.166666667 = 4.400126261

Simplifying
n + -1.166666667 = 4.400126261

Reorder the terms:
-1.166666667 + n = 4.400126261

Solving
-1.166666667 + n = 4.400126261

Solving for variable 'n'.

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

Add '1.166666667' to each side of the equation.
-1.166666667 + 1.166666667 + n = 4.400126261 + 1.166666667

Combine like terms: -1.166666667 + 1.166666667 = 0.000000000
0.000000000 + n = 4.400126261 + 1.166666667
n = 4.400126261 + 1.166666667

Combine like terms: 4.400126261 + 1.166666667 = 5.566792928
n = 5.566792928

Simplifying
n = 5.566792928


Subproblem 2
n + -1.166666667 = -4.400126261

Simplifying
n + -1.166666667 = -4.400126261

Reorder the terms:
-1.166666667 + n = -4.400126261

Solving
-1.166666667 + n = -4.400126261

Solving for variable 'n'.

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

Add '1.166666667' to each side of the equation.
-1.166666667 + 1.166666667 + n = -4.400126261 + 1.166666667

Combine like terms: -1.166666667 + 1.166666667 = 0.000000000
0.000000000 + n = -4.400126261 + 1.166666667
n = -4.400126261 + 1.166666667

Combine like terms: -4.400126261 + 1.166666667 = -3.233459594
n = -3.233459594

Simplifying
n = -3.233459594


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {5.566792928, -3.233459594}
Solution
n = {5.566792928, -3.233459594}

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