(n-5)(n-6)210=1

Solution for (n-5)(n-6)210=1 equation:

Simplifying
(n + -5)(n + -6) * 210 = 1

Reorder the terms:
(-5 + n)(n + -6) * 210 = 1

Reorder the terms:
(-5 + n)(-6 + n) * 210 = 1

Reorder the terms for easier multiplication:
210(-5 + n)(-6 + n) = 1

Multiply (-5 + n) * (-6 + n)
210(-5(-6 + n) + n(-6 + n)) = 1
210((-6 * -5 + n * -5) + n(-6 + n)) = 1
210((30 + -5n) + n(-6 + n)) = 1
210(30 + -5n + (-6 * n + n * n)) = 1
210(30 + -5n + (-6n + n2)) = 1

Combine like terms: -5n + -6n = -11n
210(30 + -11n + n2) = 1
(30 * 210 + -11n * 210 + n2 * 210) = 1
(6300 + -2310n + 210n2) = 1

Solving
6300 + -2310n + 210n2 = 1

Solving for variable 'n'.

Reorder the terms:
6300 + -1 + -2310n + 210n2 = 1 + -1

Combine like terms: 6300 + -1 = 6299
6299 + -2310n + 210n2 = 1 + -1

Combine like terms: 1 + -1 = 0
6299 + -2310n + 210n2 = 0

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

Divide each side by '210'.
29.9952381 + -11n + n2 = 0

Move the constant term to the right:

Add '-29.9952381' to each side of the equation.
29.9952381 + -11n + -29.9952381 + n2 = 0 + -29.9952381

Reorder the terms:
29.9952381 + -29.9952381 + -11n + n2 = 0 + -29.9952381

Combine like terms: 29.9952381 + -29.9952381 = 0.0000000
0.0000000 + -11n + n2 = 0 + -29.9952381
-11n + n2 = 0 + -29.9952381

Combine like terms: 0 + -29.9952381 = -29.9952381
-11n + n2 = -29.9952381

The n term is -11n.  Take half its coefficient (-5.5).
Square it (30.25) and add it to both sides.

Add '30.25' to each side of the equation.
-11n + 30.25 + n2 = -29.9952381 + 30.25

Reorder the terms:
30.25 + -11n + n2 = -29.9952381 + 30.25

Combine like terms: -29.9952381 + 30.25 = 0.2547619
30.25 + -11n + n2 = 0.2547619

Factor a perfect square on the left side:
(n + -5.5)(n + -5.5) = 0.2547619

Calculate the square root of the right side: 0.504739438

Break this problem into two subproblems by setting 
(n + -5.5) equal to 0.504739438 and -0.504739438.

Subproblem 1
n + -5.5 = 0.504739438

Simplifying
n + -5.5 = 0.504739438

Reorder the terms:
-5.5 + n = 0.504739438

Solving
-5.5 + n = 0.504739438

Solving for variable 'n'.

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

Add '5.5' to each side of the equation.
-5.5 + 5.5 + n = 0.504739438 + 5.5

Combine like terms: -5.5 + 5.5 = 0.0
0.0 + n = 0.504739438 + 5.5
n = 0.504739438 + 5.5

Combine like terms: 0.504739438 + 5.5 = 6.004739438
n = 6.004739438

Simplifying
n = 6.004739438

Subproblem 2
n + -5.5 = -0.504739438

Simplifying
n + -5.5 = -0.504739438

Reorder the terms:
-5.5 + n = -0.504739438

Solving
-5.5 + n = -0.504739438

Solving for variable 'n'.

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

Add '5.5' to each side of the equation.
-5.5 + 5.5 + n = -0.504739438 + 5.5

Combine like terms: -5.5 + 5.5 = 0.0
0.0 + n = -0.504739438 + 5.5
n = -0.504739438 + 5.5

Combine like terms: -0.504739438 + 5.5 = 4.995260562
n = 4.995260562

Simplifying
n = 4.995260562

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {6.004739438, 4.995260562}