6x^3-54x^2+84=6x(x-2)(x+7)

Solution for 6x^3-54x^2+84=6x(x-2)(x+7) equation:

Simplifying
6x3 + -54x2 + 84 = 6x(x + -2)(x + 7)

Reorder the terms:
84 + -54x2 + 6x3 = 6x(x + -2)(x + 7)

Reorder the terms:
84 + -54x2 + 6x3 = 6x(-2 + x)(x + 7)

Reorder the terms:
84 + -54x2 + 6x3 = 6x(-2 + x)(7 + x)

Multiply (-2 + x) * (7 + x)
84 + -54x2 + 6x3 = 6x(-2(7 + x) + x(7 + x))
84 + -54x2 + 6x3 = 6x((7 * -2 + x * -2) + x(7 + x))
84 + -54x2 + 6x3 = 6x((-14 + -2x) + x(7 + x))
84 + -54x2 + 6x3 = 6x(-14 + -2x + (7 * x + x * x))
84 + -54x2 + 6x3 = 6x(-14 + -2x + (7x + x2))

Combine like terms: -2x + 7x = 5x
84 + -54x2 + 6x3 = 6x(-14 + 5x + x2)
84 + -54x2 + 6x3 = (-14 * 6x + 5x * 6x + x2 * 6x)
84 + -54x2 + 6x3 = (-84x + 30x2 + 6x3)

Add '-6x3' to each side of the equation.
84 + -54x2 + 6x3 + -6x3 = -84x + 30x2 + 6x3 + -6x3

Combine like terms: 6x3 + -6x3 = 0
84 + -54x2 + 0 = -84x + 30x2 + 6x3 + -6x3
84 + -54x2 = -84x + 30x2 + 6x3 + -6x3

Combine like terms: 6x3 + -6x3 = 0
84 + -54x2 = -84x + 30x2 + 0
84 + -54x2 = -84x + 30x2

Solving
84 + -54x2 = -84x + 30x2

Solving for variable 'x'.

Reorder the terms:
84 + 84x + -54x2 + -30x2 = -84x + 30x2 + 84x + -30x2

Combine like terms: -54x2 + -30x2 = -84x2
84 + 84x + -84x2 = -84x + 30x2 + 84x + -30x2

Reorder the terms:
84 + 84x + -84x2 = -84x + 84x + 30x2 + -30x2

Combine like terms: -84x + 84x = 0
84 + 84x + -84x2 = 0 + 30x2 + -30x2
84 + 84x + -84x2 = 30x2 + -30x2

Combine like terms: 30x2 + -30x2 = 0
84 + 84x + -84x2 = 0

Factor out the Greatest Common Factor (GCF), '84'.
84(1 + x + -1x2) = 0

Ignore the factor 84.
Subproblem 1
Set the factor '(1 + x + -1x2)' equal to zero and attempt to solve:

Simplifying
1 + x + -1x2 = 0

Solving
1 + x + -1x2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-1 + -1x + x2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -1x + 1 + x2 = 0 + 1

Reorder the terms:
-1 + 1 + -1x + x2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -1x + x2 = 0 + 1
-1x + x2 = 0 + 1

Combine like terms: 0 + 1 = 1
-1x + x2 = 1

The x term is x.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1x + 0.25 + x2 = 1 + 0.25

Reorder the terms:
0.25 + -1x + x2 = 1 + 0.25

Combine like terms: 1 + 0.25 = 1.25
0.25 + -1x + x2 = 1.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 1.25

Calculate the square root of the right side: 1.118033989

Break this problem into two subproblems by setting 
(x + 0.5) equal to 1.118033989 and -1.118033989.

Subproblem 1
x + 0.5 = 1.118033989

Simplifying
x + 0.5 = 1.118033989

Reorder the terms:
0.5 + x = 1.118033989

Solving
0.5 + x = 1.118033989

Solving for variable 'x'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = 1.118033989 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 1.118033989 + -0.5
x = 1.118033989 + -0.5

Combine like terms: 1.118033989 + -0.5 = 0.618033989
x = 0.618033989

Simplifying
x = 0.618033989

Subproblem 2
x + 0.5 = -1.118033989

Simplifying
x + 0.5 = -1.118033989

Reorder the terms:
0.5 + x = -1.118033989

Solving
0.5 + x = -1.118033989

Solving for variable 'x'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = -1.118033989 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -1.118033989 + -0.5
x = -1.118033989 + -0.5

Combine like terms: -1.118033989 + -0.5 = -1.618033989
x = -1.618033989

Simplifying
x = -1.618033989

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.618033989, -1.618033989}
Solution
x = {0.618033989, -1.618033989}