3n^2-4n-489=0

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Solution for 3n^2-4n-489=0 equation: 


Simplifying
3n2 + -4n + -489 = 0

Reorder the terms:
-489 + -4n + 3n2 = 0

Solving
-489 + -4n + 3n2 = 0

Solving for variable 'n'.

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

Divide each side by '3'.
-163 + -1.333333333n + n2 = 0

Move the constant term to the right:

Add '163' to each side of the equation.
-163 + -1.333333333n + 163 + n2 = 0 + 163

Reorder the terms:
-163 + 163 + -1.333333333n + n2 = 0 + 163

Combine like terms: -163 + 163 = 0
0 + -1.333333333n + n2 = 0 + 163
-1.333333333n + n2 = 0 + 163

Combine like terms: 0 + 163 = 163
-1.333333333n + n2 = 163

The n term is -1.333333333n.  Take half its coefficient (-0.6666666665).
Square it (0.4444444442) and add it to both sides.

Add '0.4444444442' to each side of the equation.
-1.333333333n + 0.4444444442 + n2 = 163 + 0.4444444442

Reorder the terms:
0.4444444442 + -1.333333333n + n2 = 163 + 0.4444444442

Combine like terms: 163 + 0.4444444442 = 163.4444444442
0.4444444442 + -1.333333333n + n2 = 163.4444444442

Factor a perfect square on the left side:
(n + -0.6666666665)(n + -0.6666666665) = 163.4444444442

Calculate the square root of the right side: 12.784539274

Break this problem into two subproblems by setting 
(n + -0.6666666665) equal to 12.784539274 and -12.784539274.


Subproblem 1
n + -0.6666666665 = 12.784539274

Simplifying
n + -0.6666666665 = 12.784539274

Reorder the terms:
-0.6666666665 + n = 12.784539274

Solving
-0.6666666665 + n = 12.784539274

Solving for variable 'n'.

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

Add '0.6666666665' to each side of the equation.
-0.6666666665 + 0.6666666665 + n = 12.784539274 + 0.6666666665

Combine like terms: -0.6666666665 + 0.6666666665 = 0.0000000000
0.0000000000 + n = 12.784539274 + 0.6666666665
n = 12.784539274 + 0.6666666665

Combine like terms: 12.784539274 + 0.6666666665 = 13.4512059405
n = 13.4512059405

Simplifying
n = 13.4512059405


Subproblem 2
n + -0.6666666665 = -12.784539274

Simplifying
n + -0.6666666665 = -12.784539274

Reorder the terms:
-0.6666666665 + n = -12.784539274

Solving
-0.6666666665 + n = -12.784539274

Solving for variable 'n'.

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

Add '0.6666666665' to each side of the equation.
-0.6666666665 + 0.6666666665 + n = -12.784539274 + 0.6666666665

Combine like terms: -0.6666666665 + 0.6666666665 = 0.0000000000
0.0000000000 + n = -12.784539274 + 0.6666666665
n = -12.784539274 + 0.6666666665

Combine like terms: -12.784539274 + 0.6666666665 = -12.1178726075
n = -12.1178726075

Simplifying
n = -12.1178726075


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {13.4512059405, -12.1178726075}

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