(4x^2+x)(3)-(3x-2)(8x+1)=0

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Solution for (4x^2+x)(3)-(3x-2)(8x+1)=0 equation: 


Simplifying
(4x2 + x)(3) + -1(3x + -2)(8x + 1) = 0

Reorder the terms:
(x + 4x2)(3) + -1(3x + -2)(8x + 1) = 0

Reorder the terms for easier multiplication:
3(x + 4x2) + -1(3x + -2)(8x + 1) = 0
(x * 3 + 4x2 * 3) + -1(3x + -2)(8x + 1) = 0
(3x + 12x2) + -1(3x + -2)(8x + 1) = 0

Reorder the terms:
3x + 12x2 + -1(-2 + 3x)(8x + 1) = 0

Reorder the terms:
3x + 12x2 + -1(-2 + 3x)(1 + 8x) = 0

Multiply (-2 + 3x) * (1 + 8x)
3x + 12x2 + -1(-2(1 + 8x) + 3x * (1 + 8x)) = 0
3x + 12x2 + -1((1 * -2 + 8x * -2) + 3x * (1 + 8x)) = 0
3x + 12x2 + -1((-2 + -16x) + 3x * (1 + 8x)) = 0
3x + 12x2 + -1(-2 + -16x + (1 * 3x + 8x * 3x)) = 0
3x + 12x2 + -1(-2 + -16x + (3x + 24x2)) = 0

Combine like terms: -16x + 3x = -13x
3x + 12x2 + -1(-2 + -13x + 24x2) = 0
3x + 12x2 + (-2 * -1 + -13x * -1 + 24x2 * -1) = 0
3x + 12x2 + (2 + 13x + -24x2) = 0

Reorder the terms:
2 + 3x + 13x + 12x2 + -24x2 = 0

Combine like terms: 3x + 13x = 16x
2 + 16x + 12x2 + -24x2 = 0

Combine like terms: 12x2 + -24x2 = -12x2
2 + 16x + -12x2 = 0

Solving
2 + 16x + -12x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '2'.
2(1 + 8x + -6x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(1 + 8x + -6x2)' equal to zero and attempt to solve:

Simplifying
1 + 8x + -6x2 = 0

Solving
1 + 8x + -6x2 = 0

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

Divide each side by '-6'.
-0.1666666667 + -1.333333333x + x2 = 0

Move the constant term to the right:

Add '0.1666666667' to each side of the equation.
-0.1666666667 + -1.333333333x + 0.1666666667 + x2 = 0 + 0.1666666667

Reorder the terms:
-0.1666666667 + 0.1666666667 + -1.333333333x + x2 = 0 + 0.1666666667

Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000
0.0000000000 + -1.333333333x + x2 = 0 + 0.1666666667
-1.333333333x + x2 = 0 + 0.1666666667

Combine like terms: 0 + 0.1666666667 = 0.1666666667
-1.333333333x + x2 = 0.1666666667

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 = 0.1666666667 + 0.4444444442

Reorder the terms:
0.4444444442 + -1.333333333x + x2 = 0.1666666667 + 0.4444444442

Combine like terms: 0.1666666667 + 0.4444444442 = 0.6111111109
0.4444444442 + -1.333333333x + x2 = 0.6111111109

Factor a perfect square on the left side:
(x + -0.6666666665)(x + -0.6666666665) = 0.6111111109

Calculate the square root of the right side: 0.78173596

Break this problem into two subproblems by setting 
(x + -0.6666666665) equal to 0.78173596 and -0.78173596.


Subproblem 1
x + -0.6666666665 = 0.78173596

Simplifying
x + -0.6666666665 = 0.78173596

Reorder the terms:
-0.6666666665 + x = 0.78173596

Solving
-0.6666666665 + x = 0.78173596

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 = 0.78173596 + 0.6666666665

Combine like terms: -0.6666666665 + 0.6666666665 = 0.0000000000
0.0000000000 + x = 0.78173596 + 0.6666666665
x = 0.78173596 + 0.6666666665

Combine like terms: 0.78173596 + 0.6666666665 = 1.4484026265
x = 1.4484026265

Simplifying
x = 1.4484026265


Subproblem 2
x + -0.6666666665 = -0.78173596

Simplifying
x + -0.6666666665 = -0.78173596

Reorder the terms:
-0.6666666665 + x = -0.78173596

Solving
-0.6666666665 + x = -0.78173596

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 = -0.78173596 + 0.6666666665

Combine like terms: -0.6666666665 + 0.6666666665 = 0.0000000000
0.0000000000 + x = -0.78173596 + 0.6666666665
x = -0.78173596 + 0.6666666665

Combine like terms: -0.78173596 + 0.6666666665 = -0.1150692935
x = -0.1150692935

Simplifying
x = -0.1150692935


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.4484026265, -0.1150692935}
Solution
x = {1.4484026265, -0.1150692935}

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