6n^2-4n-3=0

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Solution for 6n^2-4n-3=0 equation: 


Simplifying
6n2 + -4n + -3 = 0

Reorder the terms:
-3 + -4n + 6n2 = 0

Solving
-3 + -4n + 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'.
-0.5 + -0.6666666667n + n2 = 0

Move the constant term to the right:

Add '0.5' to each side of the equation.
-0.5 + -0.6666666667n + 0.5 + n2 = 0 + 0.5

Reorder the terms:
-0.5 + 0.5 + -0.6666666667n + n2 = 0 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + -0.6666666667n + n2 = 0 + 0.5
-0.6666666667n + n2 = 0 + 0.5

Combine like terms: 0 + 0.5 = 0.5
-0.6666666667n + n2 = 0.5

The n term is -0.6666666667n.  Take half its coefficient (-0.3333333334).
Square it (0.1111111112) and add it to both sides.

Add '0.1111111112' to each side of the equation.
-0.6666666667n + 0.1111111112 + n2 = 0.5 + 0.1111111112

Reorder the terms:
0.1111111112 + -0.6666666667n + n2 = 0.5 + 0.1111111112

Combine like terms: 0.5 + 0.1111111112 = 0.6111111112
0.1111111112 + -0.6666666667n + n2 = 0.6111111112

Factor a perfect square on the left side:
(n + -0.3333333334)(n + -0.3333333334) = 0.6111111112

Calculate the square root of the right side: 0.78173596

Break this problem into two subproblems by setting 
(n + -0.3333333334) equal to 0.78173596 and -0.78173596.


Subproblem 1
n + -0.3333333334 = 0.78173596

Simplifying
n + -0.3333333334 = 0.78173596

Reorder the terms:
-0.3333333334 + n = 0.78173596

Solving
-0.3333333334 + n = 0.78173596

Solving for variable 'n'.

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

Add '0.3333333334' to each side of the equation.
-0.3333333334 + 0.3333333334 + n = 0.78173596 + 0.3333333334

Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000
0.0000000000 + n = 0.78173596 + 0.3333333334
n = 0.78173596 + 0.3333333334

Combine like terms: 0.78173596 + 0.3333333334 = 1.1150692934
n = 1.1150692934

Simplifying
n = 1.1150692934


Subproblem 2
n + -0.3333333334 = -0.78173596

Simplifying
n + -0.3333333334 = -0.78173596

Reorder the terms:
-0.3333333334 + n = -0.78173596

Solving
-0.3333333334 + n = -0.78173596

Solving for variable 'n'.

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

Add '0.3333333334' to each side of the equation.
-0.3333333334 + 0.3333333334 + n = -0.78173596 + 0.3333333334

Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000
0.0000000000 + n = -0.78173596 + 0.3333333334
n = -0.78173596 + 0.3333333334

Combine like terms: -0.78173596 + 0.3333333334 = -0.4484026266
n = -0.4484026266

Simplifying
n = -0.4484026266


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {1.1150692934, -0.4484026266}

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