(5x+2)(3x-2)-(3x-2)(3x-2)=27

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Solution for (5x+2)(3x-2)-(3x-2)(3x-2)=27 equation: 


Simplifying
(5x + 2)(3x + -2) + -1(3x + -2)(3x + -2) = 27

Reorder the terms:
(2 + 5x)(3x + -2) + -1(3x + -2)(3x + -2) = 27

Reorder the terms:
(2 + 5x)(-2 + 3x) + -1(3x + -2)(3x + -2) = 27

Multiply (2 + 5x) * (-2 + 3x)
(2(-2 + 3x) + 5x * (-2 + 3x)) + -1(3x + -2)(3x + -2) = 27
((-2 * 2 + 3x * 2) + 5x * (-2 + 3x)) + -1(3x + -2)(3x + -2) = 27
((-4 + 6x) + 5x * (-2 + 3x)) + -1(3x + -2)(3x + -2) = 27
(-4 + 6x + (-2 * 5x + 3x * 5x)) + -1(3x + -2)(3x + -2) = 27
(-4 + 6x + (-10x + 15x2)) + -1(3x + -2)(3x + -2) = 27

Combine like terms: 6x + -10x = -4x
(-4 + -4x + 15x2) + -1(3x + -2)(3x + -2) = 27

Reorder the terms:
-4 + -4x + 15x2 + -1(-2 + 3x)(3x + -2) = 27

Reorder the terms:
-4 + -4x + 15x2 + -1(-2 + 3x)(-2 + 3x) = 27

Multiply (-2 + 3x) * (-2 + 3x)
-4 + -4x + 15x2 + -1(-2(-2 + 3x) + 3x * (-2 + 3x)) = 27
-4 + -4x + 15x2 + -1((-2 * -2 + 3x * -2) + 3x * (-2 + 3x)) = 27
-4 + -4x + 15x2 + -1((4 + -6x) + 3x * (-2 + 3x)) = 27
-4 + -4x + 15x2 + -1(4 + -6x + (-2 * 3x + 3x * 3x)) = 27
-4 + -4x + 15x2 + -1(4 + -6x + (-6x + 9x2)) = 27

Combine like terms: -6x + -6x = -12x
-4 + -4x + 15x2 + -1(4 + -12x + 9x2) = 27
-4 + -4x + 15x2 + (4 * -1 + -12x * -1 + 9x2 * -1) = 27
-4 + -4x + 15x2 + (-4 + 12x + -9x2) = 27

Reorder the terms:
-4 + -4 + -4x + 12x + 15x2 + -9x2 = 27

Combine like terms: -4 + -4 = -8
-8 + -4x + 12x + 15x2 + -9x2 = 27

Combine like terms: -4x + 12x = 8x
-8 + 8x + 15x2 + -9x2 = 27

Combine like terms: 15x2 + -9x2 = 6x2
-8 + 8x + 6x2 = 27

Solving
-8 + 8x + 6x2 = 27

Solving for variable 'x'.

Reorder the terms:
-8 + -27 + 8x + 6x2 = 27 + -27

Combine like terms: -8 + -27 = -35
-35 + 8x + 6x2 = 27 + -27

Combine like terms: 27 + -27 = 0
-35 + 8x + 6x2 = 0

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

Divide each side by '6'.
-5.833333333 + 1.333333333x + x2 = 0

Move the constant term to the right:

Add '5.833333333' to each side of the equation.
-5.833333333 + 1.333333333x + 5.833333333 + x2 = 0 + 5.833333333

Reorder the terms:
-5.833333333 + 5.833333333 + 1.333333333x + x2 = 0 + 5.833333333

Combine like terms: -5.833333333 + 5.833333333 = 0.000000000
0.000000000 + 1.333333333x + x2 = 0 + 5.833333333
1.333333333x + x2 = 0 + 5.833333333

Combine like terms: 0 + 5.833333333 = 5.833333333
1.333333333x + x2 = 5.833333333

The x term is 1.333333333x.  Take half its coefficient (0.6666666665).
Square it (0.4444444442) and add it to both sides.

Add '0.4444444442' to each side of the equation.
1.333333333x + 0.4444444442 + x2 = 5.833333333 + 0.4444444442

Reorder the terms:
0.4444444442 + 1.333333333x + x2 = 5.833333333 + 0.4444444442

Combine like terms: 5.833333333 + 0.4444444442 = 6.2777777772
0.4444444442 + 1.333333333x + x2 = 6.2777777772

Factor a perfect square on the left side:
(x + 0.6666666665)(x + 0.6666666665) = 6.2777777772

Calculate the square root of the right side: 2.505549396

Break this problem into two subproblems by setting 
(x + 0.6666666665) equal to 2.505549396 and -2.505549396.


Subproblem 1
x + 0.6666666665 = 2.505549396

Simplifying
x + 0.6666666665 = 2.505549396

Reorder the terms:
0.6666666665 + x = 2.505549396

Solving
0.6666666665 + x = 2.505549396

Solving for variable 'x'.

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

Add '-0.6666666665' to each side of the equation.
0.6666666665 + -0.6666666665 + x = 2.505549396 + -0.6666666665

Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000
0.0000000000 + x = 2.505549396 + -0.6666666665
x = 2.505549396 + -0.6666666665

Combine like terms: 2.505549396 + -0.6666666665 = 1.8388827295
x = 1.8388827295

Simplifying
x = 1.8388827295


Subproblem 2
x + 0.6666666665 = -2.505549396

Simplifying
x + 0.6666666665 = -2.505549396

Reorder the terms:
0.6666666665 + x = -2.505549396

Solving
0.6666666665 + x = -2.505549396

Solving for variable 'x'.

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

Add '-0.6666666665' to each side of the equation.
0.6666666665 + -0.6666666665 + x = -2.505549396 + -0.6666666665

Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000
0.0000000000 + x = -2.505549396 + -0.6666666665
x = -2.505549396 + -0.6666666665

Combine like terms: -2.505549396 + -0.6666666665 = -3.1722160625
x = -3.1722160625

Simplifying
x = -3.1722160625


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.8388827295, -3.1722160625}

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