3n^2-2n-3=0

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


Simplifying
3n2 + -2n + -3 = 0

Reorder the terms:
-3 + -2n + 3n2 = 0

Solving
-3 + -2n + 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'.
-1 + -0.6666666667n + n2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -0.6666666667n + 1 + n2 = 0 + 1

Reorder the terms:
-1 + 1 + -0.6666666667n + n2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -0.6666666667n + n2 = 0 + 1
-0.6666666667n + n2 = 0 + 1

Combine like terms: 0 + 1 = 1
-0.6666666667n + n2 = 1

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 = 1 + 0.1111111112

Reorder the terms:
0.1111111112 + -0.6666666667n + n2 = 1 + 0.1111111112

Combine like terms: 1 + 0.1111111112 = 1.1111111112
0.1111111112 + -0.6666666667n + n2 = 1.1111111112

Factor a perfect square on the left side:
(n + -0.3333333334)(n + -0.3333333334) = 1.1111111112

Calculate the square root of the right side: 1.054092553

Break this problem into two subproblems by setting 
(n + -0.3333333334) equal to 1.054092553 and -1.054092553.


Subproblem 1
n + -0.3333333334 = 1.054092553

Simplifying
n + -0.3333333334 = 1.054092553

Reorder the terms:
-0.3333333334 + n = 1.054092553

Solving
-0.3333333334 + n = 1.054092553

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 = 1.054092553 + 0.3333333334

Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000
0.0000000000 + n = 1.054092553 + 0.3333333334
n = 1.054092553 + 0.3333333334

Combine like terms: 1.054092553 + 0.3333333334 = 1.3874258864
n = 1.3874258864

Simplifying
n = 1.3874258864


Subproblem 2
n + -0.3333333334 = -1.054092553

Simplifying
n + -0.3333333334 = -1.054092553

Reorder the terms:
-0.3333333334 + n = -1.054092553

Solving
-0.3333333334 + n = -1.054092553

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 = -1.054092553 + 0.3333333334

Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000
0.0000000000 + n = -1.054092553 + 0.3333333334
n = -1.054092553 + 0.3333333334

Combine like terms: -1.054092553 + 0.3333333334 = -0.7207592196
n = -0.7207592196

Simplifying
n = -0.7207592196


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {1.3874258864, -0.7207592196}

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