(3)(6n)-(3n+6)(n+6)=0

Solution for (3)(6n)-(3n+6)(n+6)=0 equation:

Simplifying
(3)(6n) + -1(3n + 6)(n + 6) = 0

Remove parenthesis around (6n)
3 * 6n + -1(3n + 6)(n + 6) = 0

Multiply 3 * 6
18n + -1(3n + 6)(n + 6) = 0

Reorder the terms:
18n + -1(6 + 3n)(n + 6) = 0

Reorder the terms:
18n + -1(6 + 3n)(6 + n) = 0

Multiply (6 + 3n) * (6 + n)
18n + -1(6(6 + n) + 3n * (6 + n)) = 0
18n + -1((6 * 6 + n * 6) + 3n * (6 + n)) = 0
18n + -1((36 + 6n) + 3n * (6 + n)) = 0
18n + -1(36 + 6n + (6 * 3n + n * 3n)) = 0
18n + -1(36 + 6n + (18n + 3n2)) = 0

Combine like terms: 6n + 18n = 24n
18n + -1(36 + 24n + 3n2) = 0
18n + (36 * -1 + 24n * -1 + 3n2 * -1) = 0
18n + (-36 + -24n + -3n2) = 0

Reorder the terms:
-36 + 18n + -24n + -3n2 = 0

Combine like terms: 18n + -24n = -6n
-36 + -6n + -3n2 = 0

Solving
-36 + -6n + -3n2 = 0

Solving for variable 'n'.

Factor out the Greatest Common Factor (GCF), '-3'.
-3(12 + 2n + n2) = 0

Ignore the factor -3.
Subproblem 1
Set the factor '(12 + 2n + n2)' equal to zero and attempt to solve:

Simplifying
12 + 2n + n2 = 0

Solving
12 + 2n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-12' to each side of the equation.
12 + 2n + -12 + n2 = 0 + -12

Reorder the terms:
12 + -12 + 2n + n2 = 0 + -12

Combine like terms: 12 + -12 = 0
0 + 2n + n2 = 0 + -12
2n + n2 = 0 + -12

Combine like terms: 0 + -12 = -12
2n + n2 = -12

The n term is 2n.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2n + 1 + n2 = -12 + 1

Reorder the terms:
1 + 2n + n2 = -12 + 1

Combine like terms: -12 + 1 = -11
1 + 2n + n2 = -11

Factor a perfect square on the left side:
(n + 1)(n + 1) = -11

Can't calculate square root of the right side.

The solution to this equation could not be determined.

This subproblem is being ignored because a solution could not be determined.

The solution to this equation could not be determined.