9n-3=7(n+3)2n

Solution for 9n-3=7(n+3)2n equation:

Simplifying
9n + -3 = 7(n + 3) * 2n

Reorder the terms:
-3 + 9n = 7(n + 3) * 2n

Reorder the terms:
-3 + 9n = 7(3 + n) * 2n

Reorder the terms for easier multiplication:
-3 + 9n = 7 * 2n(3 + n)

Multiply 7 * 2
-3 + 9n = 14n(3 + n)
-3 + 9n = (3 * 14n + n * 14n)
-3 + 9n = (42n + 14n2)

Solving
-3 + 9n = 42n + 14n2

Solving for variable 'n'.

Combine like terms: 9n + -42n = -33n
-3 + -33n + -14n2 = 42n + 14n2 + -42n + -14n2

Reorder the terms:
-3 + -33n + -14n2 = 42n + -42n + 14n2 + -14n2

Combine like terms: 42n + -42n = 0
-3 + -33n + -14n2 = 0 + 14n2 + -14n2
-3 + -33n + -14n2 = 14n2 + -14n2

Combine like terms: 14n2 + -14n2 = 0
-3 + -33n + -14n2 = 0

Factor out the Greatest Common Factor (GCF), '-1'.
-1(3 + 33n + 14n2) = 0

Ignore the factor -1.
Subproblem 1
Set the factor '(3 + 33n + 14n2)' equal to zero and attempt to solve:

Simplifying
3 + 33n + 14n2 = 0

Solving
3 + 33n + 14n2 = 0

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

Divide each side by '14'.
0.2142857143 + 2.357142857n + n2 = 0

Move the constant term to the right:

Add '-0.2142857143' to each side of the equation.
0.2142857143 + 2.357142857n + -0.2142857143 + n2 = 0 + -0.2142857143

Reorder the terms:
0.2142857143 + -0.2142857143 + 2.357142857n + n2 = 0 + -0.2142857143

Combine like terms: 0.2142857143 + -0.2142857143 = 0.0000000000
0.0000000000 + 2.357142857n + n2 = 0 + -0.2142857143
2.357142857n + n2 = 0 + -0.2142857143

Combine like terms: 0 + -0.2142857143 = -0.2142857143
2.357142857n + n2 = -0.2142857143

The n term is 2.357142857n.  Take half its coefficient (1.178571429).
Square it (1.389030613) and add it to both sides.

Add '1.389030613' to each side of the equation.
2.357142857n + 1.389030613 + n2 = -0.2142857143 + 1.389030613

Reorder the terms:
1.389030613 + 2.357142857n + n2 = -0.2142857143 + 1.389030613

Combine like terms: -0.2142857143 + 1.389030613 = 1.1747448987
1.389030613 + 2.357142857n + n2 = 1.1747448987

Factor a perfect square on the left side:
(n + 1.178571429)(n + 1.178571429) = 1.1747448987

Calculate the square root of the right side: 1.083856494

Break this problem into two subproblems by setting 
(n + 1.178571429) equal to 1.083856494 and -1.083856494.

Subproblem 1
n + 1.178571429 = 1.083856494

Simplifying
n + 1.178571429 = 1.083856494

Reorder the terms:
1.178571429 + n = 1.083856494

Solving
1.178571429 + n = 1.083856494

Solving for variable 'n'.

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

Add '-1.178571429' to each side of the equation.
1.178571429 + -1.178571429 + n = 1.083856494 + -1.178571429

Combine like terms: 1.178571429 + -1.178571429 = 0.000000000
0.000000000 + n = 1.083856494 + -1.178571429
n = 1.083856494 + -1.178571429

Combine like terms: 1.083856494 + -1.178571429 = -0.094714935
n = -0.094714935

Simplifying
n = -0.094714935

Subproblem 2
n + 1.178571429 = -1.083856494

Simplifying
n + 1.178571429 = -1.083856494

Reorder the terms:
1.178571429 + n = -1.083856494

Solving
1.178571429 + n = -1.083856494

Solving for variable 'n'.

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

Add '-1.178571429' to each side of the equation.
1.178571429 + -1.178571429 + n = -1.083856494 + -1.178571429

Combine like terms: 1.178571429 + -1.178571429 = 0.000000000
0.000000000 + n = -1.083856494 + -1.178571429
n = -1.083856494 + -1.178571429

Combine like terms: -1.083856494 + -1.178571429 = -2.262427923
n = -2.262427923

Simplifying
n = -2.262427923

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {-0.094714935, -2.262427923}
Solution
n = {-0.094714935, -2.262427923}