x(2x-4)+(3x-2)=(5x+2)(x-2)

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Solution for x(2x-4)+(3x-2)=(5x+2)(x-2) equation: 


Simplifying
x(2x + -4) + (3x + -2) = (5x + 2)(x + -2)

Reorder the terms:
x(-4 + 2x) + (3x + -2) = (5x + 2)(x + -2)
(-4 * x + 2x * x) + (3x + -2) = (5x + 2)(x + -2)
(-4x + 2x2) + (3x + -2) = (5x + 2)(x + -2)

Reorder the terms:
-4x + 2x2 + (-2 + 3x) = (5x + 2)(x + -2)

Remove parenthesis around (-2 + 3x)
-4x + 2x2 + -2 + 3x = (5x + 2)(x + -2)

Reorder the terms:
-2 + -4x + 3x + 2x2 = (5x + 2)(x + -2)

Combine like terms: -4x + 3x = -1x
-2 + -1x + 2x2 = (5x + 2)(x + -2)

Reorder the terms:
-2 + -1x + 2x2 = (2 + 5x)(x + -2)

Reorder the terms:
-2 + -1x + 2x2 = (2 + 5x)(-2 + x)

Multiply (2 + 5x) * (-2 + x)
-2 + -1x + 2x2 = (2(-2 + x) + 5x * (-2 + x))
-2 + -1x + 2x2 = ((-2 * 2 + x * 2) + 5x * (-2 + x))
-2 + -1x + 2x2 = ((-4 + 2x) + 5x * (-2 + x))
-2 + -1x + 2x2 = (-4 + 2x + (-2 * 5x + x * 5x))
-2 + -1x + 2x2 = (-4 + 2x + (-10x + 5x2))

Combine like terms: 2x + -10x = -8x
-2 + -1x + 2x2 = (-4 + -8x + 5x2)

Solving
-2 + -1x + 2x2 = -4 + -8x + 5x2

Solving for variable 'x'.

Reorder the terms:
-2 + 4 + -1x + 8x + 2x2 + -5x2 = -4 + -8x + 5x2 + 4 + 8x + -5x2

Combine like terms: -2 + 4 = 2
2 + -1x + 8x + 2x2 + -5x2 = -4 + -8x + 5x2 + 4 + 8x + -5x2

Combine like terms: -1x + 8x = 7x
2 + 7x + 2x2 + -5x2 = -4 + -8x + 5x2 + 4 + 8x + -5x2

Combine like terms: 2x2 + -5x2 = -3x2
2 + 7x + -3x2 = -4 + -8x + 5x2 + 4 + 8x + -5x2

Reorder the terms:
2 + 7x + -3x2 = -4 + 4 + -8x + 8x + 5x2 + -5x2

Combine like terms: -4 + 4 = 0
2 + 7x + -3x2 = 0 + -8x + 8x + 5x2 + -5x2
2 + 7x + -3x2 = -8x + 8x + 5x2 + -5x2

Combine like terms: -8x + 8x = 0
2 + 7x + -3x2 = 0 + 5x2 + -5x2
2 + 7x + -3x2 = 5x2 + -5x2

Combine like terms: 5x2 + -5x2 = 0
2 + 7x + -3x2 = 0

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

Divide each side by '-3'.
-0.6666666667 + -2.333333333x + x2 = 0

Move the constant term to the right:

Add '0.6666666667' to each side of the equation.
-0.6666666667 + -2.333333333x + 0.6666666667 + x2 = 0 + 0.6666666667

Reorder the terms:
-0.6666666667 + 0.6666666667 + -2.333333333x + x2 = 0 + 0.6666666667

Combine like terms: -0.6666666667 + 0.6666666667 = 0.0000000000
0.0000000000 + -2.333333333x + x2 = 0 + 0.6666666667
-2.333333333x + x2 = 0 + 0.6666666667

Combine like terms: 0 + 0.6666666667 = 0.6666666667
-2.333333333x + x2 = 0.6666666667

The x term is -2.333333333x.  Take half its coefficient (-1.166666667).
Square it (1.361111112) and add it to both sides.

Add '1.361111112' to each side of the equation.
-2.333333333x + 1.361111112 + x2 = 0.6666666667 + 1.361111112

Reorder the terms:
1.361111112 + -2.333333333x + x2 = 0.6666666667 + 1.361111112

Combine like terms: 0.6666666667 + 1.361111112 = 2.0277777787
1.361111112 + -2.333333333x + x2 = 2.0277777787

Factor a perfect square on the left side:
(x + -1.166666667)(x + -1.166666667) = 2.0277777787

Calculate the square root of the right side: 1.424000625

Break this problem into two subproblems by setting 
(x + -1.166666667) equal to 1.424000625 and -1.424000625.


Subproblem 1
x + -1.166666667 = 1.424000625

Simplifying
x + -1.166666667 = 1.424000625

Reorder the terms:
-1.166666667 + x = 1.424000625

Solving
-1.166666667 + x = 1.424000625

Solving for variable 'x'.

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

Add '1.166666667' to each side of the equation.
-1.166666667 + 1.166666667 + x = 1.424000625 + 1.166666667

Combine like terms: -1.166666667 + 1.166666667 = 0.000000000
0.000000000 + x = 1.424000625 + 1.166666667
x = 1.424000625 + 1.166666667

Combine like terms: 1.424000625 + 1.166666667 = 2.590667292
x = 2.590667292

Simplifying
x = 2.590667292


Subproblem 2
x + -1.166666667 = -1.424000625

Simplifying
x + -1.166666667 = -1.424000625

Reorder the terms:
-1.166666667 + x = -1.424000625

Solving
-1.166666667 + x = -1.424000625

Solving for variable 'x'.

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

Add '1.166666667' to each side of the equation.
-1.166666667 + 1.166666667 + x = -1.424000625 + 1.166666667

Combine like terms: -1.166666667 + 1.166666667 = 0.000000000
0.000000000 + x = -1.424000625 + 1.166666667
x = -1.424000625 + 1.166666667

Combine like terms: -1.424000625 + 1.166666667 = -0.257333958
x = -0.257333958

Simplifying
x = -0.257333958


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.590667292, -0.257333958}

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