6n^2-10n-16=3

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Solution for 6n^2-10n-16=3 equation: 


Simplifying
6n2 + -10n + -16 = 3

Reorder the terms:
-16 + -10n + 6n2 = 3

Solving
-16 + -10n + 6n2 = 3

Solving for variable 'n'.

Reorder the terms:
-16 + -3 + -10n + 6n2 = 3 + -3

Combine like terms: -16 + -3 = -19
-19 + -10n + 6n2 = 3 + -3

Combine like terms: 3 + -3 = 0
-19 + -10n + 6n2 = 0

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

Divide each side by '6'.
-3.166666667 + -1.666666667n + n2 = 0

Move the constant term to the right:

Add '3.166666667' to each side of the equation.
-3.166666667 + -1.666666667n + 3.166666667 + n2 = 0 + 3.166666667

Reorder the terms:
-3.166666667 + 3.166666667 + -1.666666667n + n2 = 0 + 3.166666667

Combine like terms: -3.166666667 + 3.166666667 = 0.000000000
0.000000000 + -1.666666667n + n2 = 0 + 3.166666667
-1.666666667n + n2 = 0 + 3.166666667

Combine like terms: 0 + 3.166666667 = 3.166666667
-1.666666667n + n2 = 3.166666667

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 = 3.166666667 + 0.6944444447

Reorder the terms:
0.6944444447 + -1.666666667n + n2 = 3.166666667 + 0.6944444447

Combine like terms: 3.166666667 + 0.6944444447 = 3.8611111117
0.6944444447 + -1.666666667n + n2 = 3.8611111117

Factor a perfect square on the left side:
(n + -0.8333333335)(n + -0.8333333335) = 3.8611111117

Calculate the square root of the right side: 1.964971021

Break this problem into two subproblems by setting 
(n + -0.8333333335) equal to 1.964971021 and -1.964971021.


Subproblem 1
n + -0.8333333335 = 1.964971021

Simplifying
n + -0.8333333335 = 1.964971021

Reorder the terms:
-0.8333333335 + n = 1.964971021

Solving
-0.8333333335 + n = 1.964971021

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 = 1.964971021 + 0.8333333335

Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000
0.0000000000 + n = 1.964971021 + 0.8333333335
n = 1.964971021 + 0.8333333335

Combine like terms: 1.964971021 + 0.8333333335 = 2.7983043545
n = 2.7983043545

Simplifying
n = 2.7983043545


Subproblem 2
n + -0.8333333335 = -1.964971021

Simplifying
n + -0.8333333335 = -1.964971021

Reorder the terms:
-0.8333333335 + n = -1.964971021

Solving
-0.8333333335 + n = -1.964971021

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 = -1.964971021 + 0.8333333335

Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000
0.0000000000 + n = -1.964971021 + 0.8333333335
n = -1.964971021 + 0.8333333335

Combine like terms: -1.964971021 + 0.8333333335 = -1.1316376875
n = -1.1316376875

Simplifying
n = -1.1316376875


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {2.7983043545, -1.1316376875}

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