3n(2n+1)=21

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Solution for 3n(2n+1)=21 equation: 


Simplifying
3n(2n + 1) = 21

Reorder the terms:
3n(1 + 2n) = 21
(1 * 3n + 2n * 3n) = 21
(3n + 6n2) = 21

Solving
3n + 6n2 = 21

Solving for variable 'n'.

Reorder the terms:
-21 + 3n + 6n2 = 21 + -21

Combine like terms: 21 + -21 = 0
-21 + 3n + 6n2 = 0

Factor out the Greatest Common Factor (GCF), '3'.
3(-7 + n + 2n2) = 0

Ignore the factor 3.

Subproblem 1
Set the factor '(-7 + n + 2n2)' equal to zero and attempt to solve:

Simplifying
-7 + n + 2n2 = 0

Solving
-7 + n + 2n2 = 0

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

Divide each side by '2'.
-3.5 + 0.5n + n2 = 0

Move the constant term to the right:

Add '3.5' to each side of the equation.
-3.5 + 0.5n + 3.5 + n2 = 0 + 3.5

Reorder the terms:
-3.5 + 3.5 + 0.5n + n2 = 0 + 3.5

Combine like terms: -3.5 + 3.5 = 0.0
0.0 + 0.5n + n2 = 0 + 3.5
0.5n + n2 = 0 + 3.5

Combine like terms: 0 + 3.5 = 3.5
0.5n + n2 = 3.5

The n term is n.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
0.5n + 0.25 + n2 = 3.5 + 0.25

Reorder the terms:
0.25 + 0.5n + n2 = 3.5 + 0.25

Combine like terms: 3.5 + 0.25 = 3.75
0.25 + 0.5n + n2 = 3.75

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 3.75

Calculate the square root of the right side: 1.936491673

Break this problem into two subproblems by setting 
(n + 0.5) equal to 1.936491673 and -1.936491673.


Subproblem 1
n + 0.5 = 1.936491673

Simplifying
n + 0.5 = 1.936491673

Reorder the terms:
0.5 + n = 1.936491673

Solving
0.5 + n = 1.936491673

Solving for variable 'n'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + n = 1.936491673 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + n = 1.936491673 + -0.5
n = 1.936491673 + -0.5

Combine like terms: 1.936491673 + -0.5 = 1.436491673
n = 1.436491673

Simplifying
n = 1.436491673


Subproblem 2
n + 0.5 = -1.936491673

Simplifying
n + 0.5 = -1.936491673

Reorder the terms:
0.5 + n = -1.936491673

Solving
0.5 + n = -1.936491673

Solving for variable 'n'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + n = -1.936491673 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + n = -1.936491673 + -0.5
n = -1.936491673 + -0.5

Combine like terms: -1.936491673 + -0.5 = -2.436491673
n = -2.436491673

Simplifying
n = -2.436491673


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {1.436491673, -2.436491673}
Solution
n = {1.436491673, -2.436491673}

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