6n^2-n-19=0

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Solution for 6n^2-n-19=0 equation: 


Simplifying
6n2 + -1n + -19 = 0

Reorder the terms:
-19 + -1n + 6n2 = 0

Solving
-19 + -1n + 6n2 = 0

Solving for variable 'n'.

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

Divide each side by '6'.
-3.166666667 + -0.1666666667n + n2 = 0

Move the constant term to the right:

Add '3.166666667' to each side of the equation.
-3.166666667 + -0.1666666667n + 3.166666667 + n2 = 0 + 3.166666667

Reorder the terms:
-3.166666667 + 3.166666667 + -0.1666666667n + n2 = 0 + 3.166666667

Combine like terms: -3.166666667 + 3.166666667 = 0.000000000
0.000000000 + -0.1666666667n + n2 = 0 + 3.166666667
-0.1666666667n + n2 = 0 + 3.166666667

Combine like terms: 0 + 3.166666667 = 3.166666667
-0.1666666667n + n2 = 3.166666667

The n term is -0.1666666667n.  Take half its coefficient (-0.08333333335).
Square it (0.006944444447) and add it to both sides.

Add '0.006944444447' to each side of the equation.
-0.1666666667n + 0.006944444447 + n2 = 3.166666667 + 0.006944444447

Reorder the terms:
0.006944444447 + -0.1666666667n + n2 = 3.166666667 + 0.006944444447

Combine like terms: 3.166666667 + 0.006944444447 = 3.173611111447
0.006944444447 + -0.1666666667n + n2 = 3.173611111447

Factor a perfect square on the left side:
(n + -0.08333333335)(n + -0.08333333335) = 3.173611111447

Calculate the square root of the right side: 1.781463194

Break this problem into two subproblems by setting 
(n + -0.08333333335) equal to 1.781463194 and -1.781463194.


Subproblem 1
n + -0.08333333335 = 1.781463194

Simplifying
n + -0.08333333335 = 1.781463194

Reorder the terms:
-0.08333333335 + n = 1.781463194

Solving
-0.08333333335 + n = 1.781463194

Solving for variable 'n'.

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

Add '0.08333333335' to each side of the equation.
-0.08333333335 + 0.08333333335 + n = 1.781463194 + 0.08333333335

Combine like terms: -0.08333333335 + 0.08333333335 = 0.00000000000
0.00000000000 + n = 1.781463194 + 0.08333333335
n = 1.781463194 + 0.08333333335

Combine like terms: 1.781463194 + 0.08333333335 = 1.86479652735
n = 1.86479652735

Simplifying
n = 1.86479652735


Subproblem 2
n + -0.08333333335 = -1.781463194

Simplifying
n + -0.08333333335 = -1.781463194

Reorder the terms:
-0.08333333335 + n = -1.781463194

Solving
-0.08333333335 + n = -1.781463194

Solving for variable 'n'.

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

Add '0.08333333335' to each side of the equation.
-0.08333333335 + 0.08333333335 + n = -1.781463194 + 0.08333333335

Combine like terms: -0.08333333335 + 0.08333333335 = 0.00000000000
0.00000000000 + n = -1.781463194 + 0.08333333335
n = -1.781463194 + 0.08333333335

Combine like terms: -1.781463194 + 0.08333333335 = -1.69812986065
n = -1.69812986065

Simplifying
n = -1.69812986065


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {1.86479652735, -1.69812986065}

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