3n(8n+13)=5-(13n-7)

Solution for 3n(8n+13)=5-(13n-7) equation:

Simplifying
3n(8n + 13) = 5 + -1(13n + -7)

Reorder the terms:
3n(13 + 8n) = 5 + -1(13n + -7)
(13 * 3n + 8n * 3n) = 5 + -1(13n + -7)
(39n + 24n2) = 5 + -1(13n + -7)

Reorder the terms:
39n + 24n2 = 5 + -1(-7 + 13n)
39n + 24n2 = 5 + (-7 * -1 + 13n * -1)
39n + 24n2 = 5 + (7 + -13n)

Combine like terms: 5 + 7 = 12
39n + 24n2 = 12 + -13n

Solving
39n + 24n2 = 12 + -13n

Solving for variable 'n'.

Reorder the terms:
-12 + 39n + 13n + 24n2 = 12 + -13n + -12 + 13n

Combine like terms: 39n + 13n = 52n
-12 + 52n + 24n2 = 12 + -13n + -12 + 13n

Reorder the terms:
-12 + 52n + 24n2 = 12 + -12 + -13n + 13n

Combine like terms: 12 + -12 = 0
-12 + 52n + 24n2 = 0 + -13n + 13n
-12 + 52n + 24n2 = -13n + 13n

Combine like terms: -13n + 13n = 0
-12 + 52n + 24n2 = 0

Factor out the Greatest Common Factor (GCF), '4'.
4(-3 + 13n + 6n2) = 0

Ignore the factor 4.
Subproblem 1
Set the factor '(-3 + 13n + 6n2)' equal to zero and attempt to solve:

Simplifying
-3 + 13n + 6n2 = 0

Solving
-3 + 13n + 6n2 = 0

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

Divide each side by '6'.
-0.5 + 2.166666667n + n2 = 0

Move the constant term to the right:

Add '0.5' to each side of the equation.
-0.5 + 2.166666667n + 0.5 + n2 = 0 + 0.5

Reorder the terms:
-0.5 + 0.5 + 2.166666667n + n2 = 0 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + 2.166666667n + n2 = 0 + 0.5
2.166666667n + n2 = 0 + 0.5

Combine like terms: 0 + 0.5 = 0.5
2.166666667n + n2 = 0.5

The n term is 2.166666667n.  Take half its coefficient (1.083333334).
Square it (1.173611113) and add it to both sides.

Add '1.173611113' to each side of the equation.
2.166666667n + 1.173611113 + n2 = 0.5 + 1.173611113

Reorder the terms:
1.173611113 + 2.166666667n + n2 = 0.5 + 1.173611113

Combine like terms: 0.5 + 1.173611113 = 1.673611113
1.173611113 + 2.166666667n + n2 = 1.673611113

Factor a perfect square on the left side:
(n + 1.083333334)(n + 1.083333334) = 1.673611113

Calculate the square root of the right side: 1.293681225

Break this problem into two subproblems by setting 
(n + 1.083333334) equal to 1.293681225 and -1.293681225.

Subproblem 1
n + 1.083333334 = 1.293681225

Simplifying
n + 1.083333334 = 1.293681225

Reorder the terms:
1.083333334 + n = 1.293681225

Solving
1.083333334 + n = 1.293681225

Solving for variable 'n'.

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

Add '-1.083333334' to each side of the equation.
1.083333334 + -1.083333334 + n = 1.293681225 + -1.083333334

Combine like terms: 1.083333334 + -1.083333334 = 0.000000000
0.000000000 + n = 1.293681225 + -1.083333334
n = 1.293681225 + -1.083333334

Combine like terms: 1.293681225 + -1.083333334 = 0.210347891
n = 0.210347891

Simplifying
n = 0.210347891

Subproblem 2
n + 1.083333334 = -1.293681225

Simplifying
n + 1.083333334 = -1.293681225

Reorder the terms:
1.083333334 + n = -1.293681225

Solving
1.083333334 + n = -1.293681225

Solving for variable 'n'.

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

Add '-1.083333334' to each side of the equation.
1.083333334 + -1.083333334 + n = -1.293681225 + -1.083333334

Combine like terms: 1.083333334 + -1.083333334 = 0.000000000
0.000000000 + n = -1.293681225 + -1.083333334
n = -1.293681225 + -1.083333334

Combine like terms: -1.293681225 + -1.083333334 = -2.377014559
n = -2.377014559

Simplifying
n = -2.377014559

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {0.210347891, -2.377014559}
Solution
n = {0.210347891, -2.377014559}