(x-6)(2x-6)(3)=408

Solution for (x-6)(2x-6)(3)=408 equation:

Simplifying
(x + -6)(2x + -6)(3) = 408

Reorder the terms:
(-6 + x)(2x + -6)(3) = 408

Reorder the terms:
(-6 + x)(-6 + 2x)(3) = 408

Reorder the terms for easier multiplication:
3(-6 + x)(-6 + 2x) = 408

Multiply (-6 + x) * (-6 + 2x)
3(-6(-6 + 2x) + x(-6 + 2x)) = 408
3((-6 * -6 + 2x * -6) + x(-6 + 2x)) = 408
3((36 + -12x) + x(-6 + 2x)) = 408
3(36 + -12x + (-6 * x + 2x * x)) = 408
3(36 + -12x + (-6x + 2x2)) = 408

Combine like terms: -12x + -6x = -18x
3(36 + -18x + 2x2) = 408
(36 * 3 + -18x * 3 + 2x2 * 3) = 408
(108 + -54x + 6x2) = 408

Solving
108 + -54x + 6x2 = 408

Solving for variable 'x'.

Reorder the terms:
108 + -408 + -54x + 6x2 = 408 + -408

Combine like terms: 108 + -408 = -300
-300 + -54x + 6x2 = 408 + -408

Combine like terms: 408 + -408 = 0
-300 + -54x + 6x2 = 0

Factor out the Greatest Common Factor (GCF), '6'.
6(-50 + -9x + x2) = 0

Ignore the factor 6.
Subproblem 1
Set the factor '(-50 + -9x + x2)' equal to zero and attempt to solve:

Simplifying
-50 + -9x + x2 = 0

Solving
-50 + -9x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '50' to each side of the equation.
-50 + -9x + 50 + x2 = 0 + 50

Reorder the terms:
-50 + 50 + -9x + x2 = 0 + 50

Combine like terms: -50 + 50 = 0
0 + -9x + x2 = 0 + 50
-9x + x2 = 0 + 50

Combine like terms: 0 + 50 = 50
-9x + x2 = 50

The x term is -9x.  Take half its coefficient (-4.5).
Square it (20.25) and add it to both sides.

Add '20.25' to each side of the equation.
-9x + 20.25 + x2 = 50 + 20.25

Reorder the terms:
20.25 + -9x + x2 = 50 + 20.25

Combine like terms: 50 + 20.25 = 70.25
20.25 + -9x + x2 = 70.25

Factor a perfect square on the left side:
(x + -4.5)(x + -4.5) = 70.25

Calculate the square root of the right side: 8.381527307

Break this problem into two subproblems by setting 
(x + -4.5) equal to 8.381527307 and -8.381527307.

Subproblem 1
x + -4.5 = 8.381527307

Simplifying
x + -4.5 = 8.381527307

Reorder the terms:
-4.5 + x = 8.381527307

Solving
-4.5 + x = 8.381527307

Solving for variable 'x'.

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

Add '4.5' to each side of the equation.
-4.5 + 4.5 + x = 8.381527307 + 4.5

Combine like terms: -4.5 + 4.5 = 0.0
0.0 + x = 8.381527307 + 4.5
x = 8.381527307 + 4.5

Combine like terms: 8.381527307 + 4.5 = 12.881527307
x = 12.881527307

Simplifying
x = 12.881527307

Subproblem 2
x + -4.5 = -8.381527307

Simplifying
x + -4.5 = -8.381527307

Reorder the terms:
-4.5 + x = -8.381527307

Solving
-4.5 + x = -8.381527307

Solving for variable 'x'.

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

Add '4.5' to each side of the equation.
-4.5 + 4.5 + x = -8.381527307 + 4.5

Combine like terms: -4.5 + 4.5 = 0.0
0.0 + x = -8.381527307 + 4.5
x = -8.381527307 + 4.5

Combine like terms: -8.381527307 + 4.5 = -3.881527307
x = -3.881527307

Simplifying
x = -3.881527307

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {12.881527307, -3.881527307}
Solution
x = {12.881527307, -3.881527307}