3n^2-2n+1=-3n^2+8n+11

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Solution for 3n^2-2n+1=-3n^2+8n+11 equation: 


Simplifying
3n2 + -2n + 1 = -3n2 + 8n + 11

Reorder the terms:
1 + -2n + 3n2 = -3n2 + 8n + 11

Reorder the terms:
1 + -2n + 3n2 = 11 + 8n + -3n2

Solving
1 + -2n + 3n2 = 11 + 8n + -3n2

Solving for variable 'n'.

Reorder the terms:
1 + -11 + -2n + -8n + 3n2 + 3n2 = 11 + 8n + -3n2 + -11 + -8n + 3n2

Combine like terms: 1 + -11 = -10
-10 + -2n + -8n + 3n2 + 3n2 = 11 + 8n + -3n2 + -11 + -8n + 3n2

Combine like terms: -2n + -8n = -10n
-10 + -10n + 3n2 + 3n2 = 11 + 8n + -3n2 + -11 + -8n + 3n2

Combine like terms: 3n2 + 3n2 = 6n2
-10 + -10n + 6n2 = 11 + 8n + -3n2 + -11 + -8n + 3n2

Reorder the terms:
-10 + -10n + 6n2 = 11 + -11 + 8n + -8n + -3n2 + 3n2

Combine like terms: 11 + -11 = 0
-10 + -10n + 6n2 = 0 + 8n + -8n + -3n2 + 3n2
-10 + -10n + 6n2 = 8n + -8n + -3n2 + 3n2

Combine like terms: 8n + -8n = 0
-10 + -10n + 6n2 = 0 + -3n2 + 3n2
-10 + -10n + 6n2 = -3n2 + 3n2

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

Factor out the Greatest Common Factor (GCF), '2'.
2(-5 + -5n + 3n2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-5 + -5n + 3n2)' equal to zero and attempt to solve:

Simplifying
-5 + -5n + 3n2 = 0

Solving
-5 + -5n + 3n2 = 0

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

Divide each side by '3'.
-1.666666667 + -1.666666667n + n2 = 0

Move the constant term to the right:

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

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

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

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

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 = 1.666666667 + 0.6944444447

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

Combine like terms: 1.666666667 + 0.6944444447 = 2.3611111117
0.6944444447 + -1.666666667n + n2 = 2.3611111117

Factor a perfect square on the left side:
(n + -0.8333333335)(n + -0.8333333335) = 2.3611111117

Calculate the square root of the right side: 1.536590743

Break this problem into two subproblems by setting 
(n + -0.8333333335) equal to 1.536590743 and -1.536590743.


Subproblem 1
n + -0.8333333335 = 1.536590743

Simplifying
n + -0.8333333335 = 1.536590743

Reorder the terms:
-0.8333333335 + n = 1.536590743

Solving
-0.8333333335 + n = 1.536590743

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 = 1.536590743 + 0.8333333335

Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000
0.0000000000 + n = 1.536590743 + 0.8333333335
n = 1.536590743 + 0.8333333335

Combine like terms: 1.536590743 + 0.8333333335 = 2.3699240765
n = 2.3699240765

Simplifying
n = 2.3699240765


Subproblem 2
n + -0.8333333335 = -1.536590743

Simplifying
n + -0.8333333335 = -1.536590743

Reorder the terms:
-0.8333333335 + n = -1.536590743

Solving
-0.8333333335 + n = -1.536590743

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 = -1.536590743 + 0.8333333335

Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000
0.0000000000 + n = -1.536590743 + 0.8333333335
n = -1.536590743 + 0.8333333335

Combine like terms: -1.536590743 + 0.8333333335 = -0.7032574095
n = -0.7032574095

Simplifying
n = -0.7032574095


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {2.3699240765, -0.7032574095}
Solution
n = {2.3699240765, -0.7032574095}

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