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

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

Simplifying
3(x + 2) = 2[x(6 + -1x)]

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

Solving
6 + 3x = 12x + -2x2

Solving for variable 'x'.

Combine like terms: 3x + -12x = -9x
6 + -9x + 2x2 = 12x + -2x2 + -12x + 2x2

Reorder the terms:
6 + -9x + 2x2 = 12x + -12x + -2x2 + 2x2

Combine like terms: 12x + -12x = 0
6 + -9x + 2x2 = 0 + -2x2 + 2x2
6 + -9x + 2x2 = -2x2 + 2x2

Combine like terms: -2x2 + 2x2 = 0
6 + -9x + 2x2 = 0

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

Divide each side by '2'.
3 + -4.5x + x2 = 0

Move the constant term to the right:

Add '-3' to each side of the equation.
3 + -4.5x + -3 + x2 = 0 + -3

Reorder the terms:
3 + -3 + -4.5x + x2 = 0 + -3

Combine like terms: 3 + -3 = 0
0 + -4.5x + x2 = 0 + -3
-4.5x + x2 = 0 + -3

Combine like terms: 0 + -3 = -3
-4.5x + x2 = -3

The x term is -4.5x.  Take half its coefficient (-2.25).
Square it (5.0625) and add it to both sides.

Add '5.0625' to each side of the equation.
-4.5x + 5.0625 + x2 = -3 + 5.0625

Reorder the terms:
5.0625 + -4.5x + x2 = -3 + 5.0625

Combine like terms: -3 + 5.0625 = 2.0625
5.0625 + -4.5x + x2 = 2.0625

Factor a perfect square on the left side:
(x + -2.25)(x + -2.25) = 2.0625

Calculate the square root of the right side: 1.436140662

Break this problem into two subproblems by setting 
(x + -2.25) equal to 1.436140662 and -1.436140662.

Subproblem 1
x + -2.25 = 1.436140662

Simplifying
x + -2.25 = 1.436140662

Reorder the terms:
-2.25 + x = 1.436140662

Solving
-2.25 + x = 1.436140662

Solving for variable 'x'.

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

Add '2.25' to each side of the equation.
-2.25 + 2.25 + x = 1.436140662 + 2.25

Combine like terms: -2.25 + 2.25 = 0.00
0.00 + x = 1.436140662 + 2.25
x = 1.436140662 + 2.25

Combine like terms: 1.436140662 + 2.25 = 3.686140662
x = 3.686140662

Simplifying
x = 3.686140662

Subproblem 2
x + -2.25 = -1.436140662

Simplifying
x + -2.25 = -1.436140662

Reorder the terms:
-2.25 + x = -1.436140662

Solving
-2.25 + x = -1.436140662

Solving for variable 'x'.

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

Add '2.25' to each side of the equation.
-2.25 + 2.25 + x = -1.436140662 + 2.25

Combine like terms: -2.25 + 2.25 = 0.00
0.00 + x = -1.436140662 + 2.25
x = -1.436140662 + 2.25

Combine like terms: -1.436140662 + 2.25 = 0.813859338
x = 0.813859338

Simplifying
x = 0.813859338

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.686140662, 0.813859338}