6x^2[(3x-4)+(4x+2)]=x

Solution for 6x^2[(3x-4)+(4x+2)]=x equation:

Simplifying
6x2[(3x + -4) + (4x + 2)] = x

Reorder the terms:
6x2[(-4 + 3x) + (4x + 2)] = x

Remove parenthesis around (-4 + 3x)
6x2[-4 + 3x + (4x + 2)] = x

Reorder the terms:
6x2[-4 + 3x + (2 + 4x)] = x

Remove parenthesis around (2 + 4x)
6x2[-4 + 3x + 2 + 4x] = x

Reorder the terms:
6x2[-4 + 2 + 3x + 4x] = x

Combine like terms: -4 + 2 = -2
6x2[-2 + 3x + 4x] = x

Combine like terms: 3x + 4x = 7x
6x2[-2 + 7x] = x
[-2 * 6x2 + 7x * 6x2] = x
[-12x2 + 42x3] = x

Solving
-12x2 + 42x3 = x

Solving for variable 'x'.

Reorder the terms:
-1x + -12x2 + 42x3 = x + -1x

Combine like terms: x + -1x = 0
-1x + -12x2 + 42x3 = 0

Factor out the Greatest Common Factor (GCF), 'x'.
x(-1 + -12x + 42x2) = 0

Subproblem 1
Set the factor 'x' equal to zero and attempt to solve:

Simplifying
x = 0

Solving
x = 0

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

Simplifying
x = 0
Subproblem 2
Set the factor '(-1 + -12x + 42x2)' equal to zero and attempt to solve:

Simplifying
-1 + -12x + 42x2 = 0

Solving
-1 + -12x + 42x2 = 0

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

Divide each side by '42'.
-0.02380952381 + -0.2857142857x + x2 = 0

Move the constant term to the right:

Add '0.02380952381' to each side of the equation.
-0.02380952381 + -0.2857142857x + 0.02380952381 + x2 = 0 + 0.02380952381

Reorder the terms:
-0.02380952381 + 0.02380952381 + -0.2857142857x + x2 = 0 + 0.02380952381

Combine like terms: -0.02380952381 + 0.02380952381 = 0.00000000000
0.00000000000 + -0.2857142857x + x2 = 0 + 0.02380952381
-0.2857142857x + x2 = 0 + 0.02380952381

Combine like terms: 0 + 0.02380952381 = 0.02380952381
-0.2857142857x + x2 = 0.02380952381

The x term is -0.2857142857x.  Take half its coefficient (-0.1428571429).
Square it (0.02040816328) and add it to both sides.

Add '0.02040816328' to each side of the equation.
-0.2857142857x + 0.02040816328 + x2 = 0.02380952381 + 0.02040816328

Reorder the terms:
0.02040816328 + -0.2857142857x + x2 = 0.02380952381 + 0.02040816328

Combine like terms: 0.02380952381 + 0.02040816328 = 0.04421768709
0.02040816328 + -0.2857142857x + x2 = 0.04421768709

Factor a perfect square on the left side:
(x + -0.1428571429)(x + -0.1428571429) = 0.04421768709

Calculate the square root of the right side: 0.210280021

Break this problem into two subproblems by setting 
(x + -0.1428571429) equal to 0.210280021 and -0.210280021.

Subproblem 1
x + -0.1428571429 = 0.210280021

Simplifying
x + -0.1428571429 = 0.210280021

Reorder the terms:
-0.1428571429 + x = 0.210280021

Solving
-0.1428571429 + x = 0.210280021

Solving for variable 'x'.

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

Add '0.1428571429' to each side of the equation.
-0.1428571429 + 0.1428571429 + x = 0.210280021 + 0.1428571429

Combine like terms: -0.1428571429 + 0.1428571429 = 0.0000000000
0.0000000000 + x = 0.210280021 + 0.1428571429
x = 0.210280021 + 0.1428571429

Combine like terms: 0.210280021 + 0.1428571429 = 0.3531371639
x = 0.3531371639

Simplifying
x = 0.3531371639

Subproblem 2
x + -0.1428571429 = -0.210280021

Simplifying
x + -0.1428571429 = -0.210280021

Reorder the terms:
-0.1428571429 + x = -0.210280021

Solving
-0.1428571429 + x = -0.210280021

Solving for variable 'x'.

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

Add '0.1428571429' to each side of the equation.
-0.1428571429 + 0.1428571429 + x = -0.210280021 + 0.1428571429

Combine like terms: -0.1428571429 + 0.1428571429 = 0.0000000000
0.0000000000 + x = -0.210280021 + 0.1428571429
x = -0.210280021 + 0.1428571429

Combine like terms: -0.210280021 + 0.1428571429 = -0.0674228781
x = -0.0674228781

Simplifying
x = -0.0674228781

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.3531371639, -0.0674228781}
Solution
x = {0, 0.3531371639, -0.0674228781}