-7n(1-2n)=7(1-3n)

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Solution for -7n(1-2n)=7(1-3n) equation: 


Simplifying
-7n(1 + -2n) = 7(1 + -3n)
(1 * -7n + -2n * -7n) = 7(1 + -3n)
(-7n + 14n2) = 7(1 + -3n)
-7n + 14n2 = (1 * 7 + -3n * 7)
-7n + 14n2 = (7 + -21n)

Solving
-7n + 14n2 = 7 + -21n

Solving for variable 'n'.

Reorder the terms:
-7 + -7n + 21n + 14n2 = 7 + -21n + -7 + 21n

Combine like terms: -7n + 21n = 14n
-7 + 14n + 14n2 = 7 + -21n + -7 + 21n

Reorder the terms:
-7 + 14n + 14n2 = 7 + -7 + -21n + 21n

Combine like terms: 7 + -7 = 0
-7 + 14n + 14n2 = 0 + -21n + 21n
-7 + 14n + 14n2 = -21n + 21n

Combine like terms: -21n + 21n = 0
-7 + 14n + 14n2 = 0

Factor out the Greatest Common Factor (GCF), '7'.
7(-1 + 2n + 2n2) = 0

Ignore the factor 7.

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

Simplifying
-1 + 2n + 2n2 = 0

Solving
-1 + 2n + 2n2 = 0

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

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

Move the constant term to the right:

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

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

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

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

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.
 + 0.25 + n2 = 0.5 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + n2 = 0.5 + 0.25

Combine like terms: 0.5 + 0.25 = 0.75
1.25 + n2 = 0.75

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 0.75

Calculate the square root of the right side: 0.866025404

Break this problem into two subproblems by setting 
(n + 0.5) equal to 0.866025404 and -0.866025404.


Subproblem 1
n + 0.5 = 0.866025404

Simplifying
n + 0.5 = 0.866025404

Reorder the terms:
0.5 + n = 0.866025404

Solving
0.5 + n = 0.866025404

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 = 0.866025404 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + n = 0.866025404 + -0.5
n = 0.866025404 + -0.5

Combine like terms: 0.866025404 + -0.5 = 0.366025404
n = 0.366025404

Simplifying
n = 0.366025404


Subproblem 2
n + 0.5 = -0.866025404

Simplifying
n + 0.5 = -0.866025404

Reorder the terms:
0.5 + n = -0.866025404

Solving
0.5 + n = -0.866025404

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 = -0.866025404 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + n = -0.866025404 + -0.5
n = -0.866025404 + -0.5

Combine like terms: -0.866025404 + -0.5 = -1.366025404
n = -1.366025404

Simplifying
n = -1.366025404


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {0.366025404, -1.366025404}
Solution
n = {0.366025404, -1.366025404}

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