2(x+4)=3x(x-2)

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Solution for 2(x+4)=3x(x-2) equation: 


Simplifying
2(x + 4) = 3x(x + -2)

Reorder the terms:
2(4 + x) = 3x(x + -2)
(4 * 2 + x * 2) = 3x(x + -2)
(8 + 2x) = 3x(x + -2)

Reorder the terms:
8 + 2x = 3x(-2 + x)
8 + 2x = (-2 * 3x + x * 3x)
8 + 2x = (-6x + 3x2)

Solving
8 + 2x = -6x + 3x2

Solving for variable 'x'.

Combine like terms: 2x + 6x = 8x
8 + 8x + -3x2 = -6x + 3x2 + 6x + -3x2

Reorder the terms:
8 + 8x + -3x2 = -6x + 6x + 3x2 + -3x2

Combine like terms: -6x + 6x = 0
8 + 8x + -3x2 = 0 + 3x2 + -3x2
8 + 8x + -3x2 = 3x2 + -3x2

Combine like terms: 3x2 + -3x2 = 0
8 + 8x + -3x2 = 0

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

Divide each side by '-3'.
-2.666666667 + -2.666666667x + x2 = 0

Move the constant term to the right:

Add '2.666666667' to each side of the equation.
-2.666666667 + -2.666666667x + 2.666666667 + x2 = 0 + 2.666666667

Reorder the terms:
-2.666666667 + 2.666666667 + -2.666666667x + x2 = 0 + 2.666666667

Combine like terms: -2.666666667 + 2.666666667 = 0.000000000
0.000000000 + -2.666666667x + x2 = 0 + 2.666666667
-2.666666667x + x2 = 0 + 2.666666667

Combine like terms: 0 + 2.666666667 = 2.666666667
-2.666666667x + x2 = 2.666666667

The x term is -2.666666667x.  Take half its coefficient (-1.333333334).
Square it (1.777777780) and add it to both sides.

Add '1.777777780' to each side of the equation.
-2.666666667x + 1.777777780 + x2 = 2.666666667 + 1.777777780

Reorder the terms:
1.777777780 + -2.666666667x + x2 = 2.666666667 + 1.777777780

Combine like terms: 2.666666667 + 1.777777780 = 4.444444447
1.777777780 + -2.666666667x + x2 = 4.444444447

Factor a perfect square on the left side:
(x + -1.333333334)(x + -1.333333334) = 4.444444447

Calculate the square root of the right side: 2.108185107

Break this problem into two subproblems by setting 
(x + -1.333333334) equal to 2.108185107 and -2.108185107.


Subproblem 1
x + -1.333333334 = 2.108185107

Simplifying
x + -1.333333334 = 2.108185107

Reorder the terms:
-1.333333334 + x = 2.108185107

Solving
-1.333333334 + x = 2.108185107

Solving for variable 'x'.

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

Add '1.333333334' to each side of the equation.
-1.333333334 + 1.333333334 + x = 2.108185107 + 1.333333334

Combine like terms: -1.333333334 + 1.333333334 = 0.000000000
0.000000000 + x = 2.108185107 + 1.333333334
x = 2.108185107 + 1.333333334

Combine like terms: 2.108185107 + 1.333333334 = 3.441518441
x = 3.441518441

Simplifying
x = 3.441518441


Subproblem 2
x + -1.333333334 = -2.108185107

Simplifying
x + -1.333333334 = -2.108185107

Reorder the terms:
-1.333333334 + x = -2.108185107

Solving
-1.333333334 + x = -2.108185107

Solving for variable 'x'.

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

Add '1.333333334' to each side of the equation.
-1.333333334 + 1.333333334 + x = -2.108185107 + 1.333333334

Combine like terms: -1.333333334 + 1.333333334 = 0.000000000
0.000000000 + x = -2.108185107 + 1.333333334
x = -2.108185107 + 1.333333334

Combine like terms: -2.108185107 + 1.333333334 = -0.774851773
x = -0.774851773

Simplifying
x = -0.774851773


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.441518441, -0.774851773}

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