3x(2x-1)=12+x

Solution for 3x(2x-1)=12+x equation:

Simplifying
3x(2x + -1) = 12 + x

Reorder the terms:
3x(-1 + 2x) = 12 + x
(-1 * 3x + 2x * 3x) = 12 + x
(-3x + 6x2) = 12 + x

Solving
-3x + 6x2 = 12 + x

Solving for variable 'x'.

Reorder the terms:
-12 + -3x + -1x + 6x2 = 12 + x + -12 + -1x

Combine like terms: -3x + -1x = -4x
-12 + -4x + 6x2 = 12 + x + -12 + -1x

Reorder the terms:
-12 + -4x + 6x2 = 12 + -12 + x + -1x

Combine like terms: 12 + -12 = 0
-12 + -4x + 6x2 = 0 + x + -1x
-12 + -4x + 6x2 = x + -1x

Combine like terms: x + -1x = 0
-12 + -4x + 6x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-6 + -2x + 3x2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-6 + -2x + 3x2)' equal to zero and attempt to solve:

Simplifying
-6 + -2x + 3x2 = 0

Solving
-6 + -2x + 3x2 = 0

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

Divide each side by '3'.
-2 + -0.6666666667x + x2 = 0

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + -0.6666666667x + 2 + x2 = 0 + 2

Reorder the terms:
-2 + 2 + -0.6666666667x + x2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + -0.6666666667x + x2 = 0 + 2
-0.6666666667x + x2 = 0 + 2

Combine like terms: 0 + 2 = 2
-0.6666666667x + x2 = 2

The x term is -0.6666666667x.  Take half its coefficient (-0.3333333334).
Square it (0.1111111112) and add it to both sides.

Add '0.1111111112' to each side of the equation.
-0.6666666667x + 0.1111111112 + x2 = 2 + 0.1111111112

Reorder the terms:
0.1111111112 + -0.6666666667x + x2 = 2 + 0.1111111112

Combine like terms: 2 + 0.1111111112 = 2.1111111112
0.1111111112 + -0.6666666667x + x2 = 2.1111111112

Factor a perfect square on the left side:
(x + -0.3333333334)(x + -0.3333333334) = 2.1111111112

Calculate the square root of the right side: 1.452966315

Break this problem into two subproblems by setting 
(x + -0.3333333334) equal to 1.452966315 and -1.452966315.

Subproblem 1
x + -0.3333333334 = 1.452966315

Simplifying
x + -0.3333333334 = 1.452966315

Reorder the terms:
-0.3333333334 + x = 1.452966315

Solving
-0.3333333334 + x = 1.452966315

Solving for variable 'x'.

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

Add '0.3333333334' to each side of the equation.
-0.3333333334 + 0.3333333334 + x = 1.452966315 + 0.3333333334

Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000
0.0000000000 + x = 1.452966315 + 0.3333333334
x = 1.452966315 + 0.3333333334

Combine like terms: 1.452966315 + 0.3333333334 = 1.7862996484
x = 1.7862996484

Simplifying
x = 1.7862996484

Subproblem 2
x + -0.3333333334 = -1.452966315

Simplifying
x + -0.3333333334 = -1.452966315

Reorder the terms:
-0.3333333334 + x = -1.452966315

Solving
-0.3333333334 + x = -1.452966315

Solving for variable 'x'.

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

Add '0.3333333334' to each side of the equation.
-0.3333333334 + 0.3333333334 + x = -1.452966315 + 0.3333333334

Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000
0.0000000000 + x = -1.452966315 + 0.3333333334
x = -1.452966315 + 0.3333333334

Combine like terms: -1.452966315 + 0.3333333334 = -1.1196329816
x = -1.1196329816

Simplifying
x = -1.1196329816

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.7862996484, -1.1196329816}
Solution
x = {1.7862996484, -1.1196329816}