6n^2-10n-13=0

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Solution for 6n^2-10n-13=0 equation: 


Simplifying
6n2 + -10n + -13 = 0

Reorder the terms:
-13 + -10n + 6n2 = 0

Solving
-13 + -10n + 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'.
-2.166666667 + -1.666666667n + n2 = 0

Move the constant term to the right:

Add '2.166666667' to each side of the equation.
-2.166666667 + -1.666666667n + 2.166666667 + n2 = 0 + 2.166666667

Reorder the terms:
-2.166666667 + 2.166666667 + -1.666666667n + n2 = 0 + 2.166666667

Combine like terms: -2.166666667 + 2.166666667 = 0.000000000
0.000000000 + -1.666666667n + n2 = 0 + 2.166666667
-1.666666667n + n2 = 0 + 2.166666667

Combine like terms: 0 + 2.166666667 = 2.166666667
-1.666666667n + n2 = 2.166666667

The n term is -1.666666667n.  Take half its coefficient (-0.8333333335).
Square it (0.6944444447) and add it to both sides.

Add '0.6944444447' to each side of the equation.
-1.666666667n + 0.6944444447 + n2 = 2.166666667 + 0.6944444447

Reorder the terms:
0.6944444447 + -1.666666667n + n2 = 2.166666667 + 0.6944444447

Combine like terms: 2.166666667 + 0.6944444447 = 2.8611111117
0.6944444447 + -1.666666667n + n2 = 2.8611111117

Factor a perfect square on the left side:
(n + -0.8333333335)(n + -0.8333333335) = 2.8611111117

Calculate the square root of the right side: 1.691481928

Break this problem into two subproblems by setting 
(n + -0.8333333335) equal to 1.691481928 and -1.691481928.


Subproblem 1
n + -0.8333333335 = 1.691481928

Simplifying
n + -0.8333333335 = 1.691481928

Reorder the terms:
-0.8333333335 + n = 1.691481928

Solving
-0.8333333335 + n = 1.691481928

Solving for variable 'n'.

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

Add '0.8333333335' to each side of the equation.
-0.8333333335 + 0.8333333335 + n = 1.691481928 + 0.8333333335

Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000
0.0000000000 + n = 1.691481928 + 0.8333333335
n = 1.691481928 + 0.8333333335

Combine like terms: 1.691481928 + 0.8333333335 = 2.5248152615
n = 2.5248152615

Simplifying
n = 2.5248152615


Subproblem 2
n + -0.8333333335 = -1.691481928

Simplifying
n + -0.8333333335 = -1.691481928

Reorder the terms:
-0.8333333335 + n = -1.691481928

Solving
-0.8333333335 + n = -1.691481928

Solving for variable 'n'.

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

Add '0.8333333335' to each side of the equation.
-0.8333333335 + 0.8333333335 + n = -1.691481928 + 0.8333333335

Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000
0.0000000000 + n = -1.691481928 + 0.8333333335
n = -1.691481928 + 0.8333333335

Combine like terms: -1.691481928 + 0.8333333335 = -0.8581485945
n = -0.8581485945

Simplifying
n = -0.8581485945


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {2.5248152615, -0.8581485945}

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