n=(5n+1)(7n-1)

Solution for n=(5n+1)(7n-1) equation:

Simplifying
n = (5n + 1)(7n + -1)

Reorder the terms:
n = (1 + 5n)(7n + -1)

Reorder the terms:
n = (1 + 5n)(-1 + 7n)

Multiply (1 + 5n) * (-1 + 7n)
n = (1(-1 + 7n) + 5n * (-1 + 7n))
n = ((-1 * 1 + 7n * 1) + 5n * (-1 + 7n))
n = ((-1 + 7n) + 5n * (-1 + 7n))
n = (-1 + 7n + (-1 * 5n + 7n * 5n))
n = (-1 + 7n + (-5n + 35n2))

Combine like terms: 7n + -5n = 2n
n = (-1 + 2n + 35n2)

Solving
n = -1 + 2n + 35n2

Solving for variable 'n'.

Reorder the terms:
1 + n + -2n + -35n2 = -1 + 2n + 35n2 + 1 + -2n + -35n2

Combine like terms: n + -2n = -1n
1 + -1n + -35n2 = -1 + 2n + 35n2 + 1 + -2n + -35n2

Reorder the terms:
1 + -1n + -35n2 = -1 + 1 + 2n + -2n + 35n2 + -35n2

Combine like terms: -1 + 1 = 0
1 + -1n + -35n2 = 0 + 2n + -2n + 35n2 + -35n2
1 + -1n + -35n2 = 2n + -2n + 35n2 + -35n2

Combine like terms: 2n + -2n = 0
1 + -1n + -35n2 = 0 + 35n2 + -35n2
1 + -1n + -35n2 = 35n2 + -35n2

Combine like terms: 35n2 + -35n2 = 0
1 + -1n + -35n2 = 0

Begin completing the square.  Divide all terms by
-35 the coefficient of the squared term: 

Divide each side by '-35'.
-0.02857142857 + 0.02857142857n + n2 = 0

Move the constant term to the right:

Add '0.02857142857' to each side of the equation.
-0.02857142857 + 0.02857142857n + 0.02857142857 + n2 = 0 + 0.02857142857

Reorder the terms:
-0.02857142857 + 0.02857142857 + 0.02857142857n + n2 = 0 + 0.02857142857

Combine like terms: -0.02857142857 + 0.02857142857 = 0.00000000000
0.00000000000 + 0.02857142857n + n2 = 0 + 0.02857142857
0.02857142857n + n2 = 0 + 0.02857142857

Combine like terms: 0 + 0.02857142857 = 0.02857142857
0.02857142857n + n2 = 0.02857142857

The n term is 0.02857142857n.  Take half its coefficient (0.01428571429).
Square it (0.0002040816328) and add it to both sides.

Add '0.0002040816328' to each side of the equation.
0.02857142857n + 0.0002040816328 + n2 = 0.02857142857 + 0.0002040816328

Reorder the terms:
0.0002040816328 + 0.02857142857n + n2 = 0.02857142857 + 0.0002040816328

Combine like terms: 0.02857142857 + 0.0002040816328 = 0.0287755102028
0.0002040816328 + 0.02857142857n + n2 = 0.0287755102028

Factor a perfect square on the left side:
(n + 0.01428571429)(n + 0.01428571429) = 0.0287755102028

Calculate the square root of the right side: 0.169633458

Break this problem into two subproblems by setting 
(n + 0.01428571429) equal to 0.169633458 and -0.169633458.

Subproblem 1
n + 0.01428571429 = 0.169633458

Simplifying
n + 0.01428571429 = 0.169633458

Reorder the terms:
0.01428571429 + n = 0.169633458

Solving
0.01428571429 + n = 0.169633458

Solving for variable 'n'.

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

Add '-0.01428571429' to each side of the equation.
0.01428571429 + -0.01428571429 + n = 0.169633458 + -0.01428571429

Combine like terms: 0.01428571429 + -0.01428571429 = 0.00000000000
0.00000000000 + n = 0.169633458 + -0.01428571429
n = 0.169633458 + -0.01428571429

Combine like terms: 0.169633458 + -0.01428571429 = 0.15534774371
n = 0.15534774371

Simplifying
n = 0.15534774371

Subproblem 2
n + 0.01428571429 = -0.169633458

Simplifying
n + 0.01428571429 = -0.169633458

Reorder the terms:
0.01428571429 + n = -0.169633458

Solving
0.01428571429 + n = -0.169633458

Solving for variable 'n'.

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

Add '-0.01428571429' to each side of the equation.
0.01428571429 + -0.01428571429 + n = -0.169633458 + -0.01428571429

Combine like terms: 0.01428571429 + -0.01428571429 = 0.00000000000
0.00000000000 + n = -0.169633458 + -0.01428571429
n = -0.169633458 + -0.01428571429

Combine like terms: -0.169633458 + -0.01428571429 = -0.18391917229
n = -0.18391917229

Simplifying
n = -0.18391917229

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {0.15534774371, -0.18391917229}