6x(3x+8)=16

Solution for 6x(3x+8)=16 equation:

Simplifying
6x(3x + 8) = 16

Reorder the terms:
6x(8 + 3x) = 16
(8 * 6x + 3x * 6x) = 16
(48x + 18x2) = 16

Solving
48x + 18x2 = 16

Solving for variable 'x'.

Reorder the terms:
-16 + 48x + 18x2 = 16 + -16

Combine like terms: 16 + -16 = 0
-16 + 48x + 18x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-8 + 24x + 9x2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-8 + 24x + 9x2)' equal to zero and attempt to solve:

Simplifying
-8 + 24x + 9x2 = 0

Solving
-8 + 24x + 9x2 = 0

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

Divide each side by '9'.
-0.8888888889 + 2.666666667x + x2 = 0

Move the constant term to the right:

Add '0.8888888889' to each side of the equation.
-0.8888888889 + 2.666666667x + 0.8888888889 + x2 = 0 + 0.8888888889

Reorder the terms:
-0.8888888889 + 0.8888888889 + 2.666666667x + x2 = 0 + 0.8888888889

Combine like terms: -0.8888888889 + 0.8888888889 = 0.0000000000
0.0000000000 + 2.666666667x + x2 = 0 + 0.8888888889
2.666666667x + x2 = 0 + 0.8888888889

Combine like terms: 0 + 0.8888888889 = 0.8888888889
2.666666667x + x2 = 0.8888888889

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 = 0.8888888889 + 1.777777780

Reorder the terms:
1.777777780 + 2.666666667x + x2 = 0.8888888889 + 1.777777780

Combine like terms: 0.8888888889 + 1.777777780 = 2.6666666689
1.777777780 + 2.666666667x + x2 = 2.6666666689

Factor a perfect square on the left side:
(x + 1.333333334)(x + 1.333333334) = 2.6666666689

Calculate the square root of the right side: 1.632993163

Break this problem into two subproblems by setting 
(x + 1.333333334) equal to 1.632993163 and -1.632993163.

Subproblem 1
x + 1.333333334 = 1.632993163

Simplifying
x + 1.333333334 = 1.632993163

Reorder the terms:
1.333333334 + x = 1.632993163

Solving
1.333333334 + x = 1.632993163

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 = 1.632993163 + -1.333333334

Combine like terms: 1.333333334 + -1.333333334 = 0.000000000
0.000000000 + x = 1.632993163 + -1.333333334
x = 1.632993163 + -1.333333334

Combine like terms: 1.632993163 + -1.333333334 = 0.299659829
x = 0.299659829

Simplifying
x = 0.299659829

Subproblem 2
x + 1.333333334 = -1.632993163

Simplifying
x + 1.333333334 = -1.632993163

Reorder the terms:
1.333333334 + x = -1.632993163

Solving
1.333333334 + x = -1.632993163

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 = -1.632993163 + -1.333333334

Combine like terms: 1.333333334 + -1.333333334 = 0.000000000
0.000000000 + x = -1.632993163 + -1.333333334
x = -1.632993163 + -1.333333334

Combine like terms: -1.632993163 + -1.333333334 = -2.966326497
x = -2.966326497

Simplifying
x = -2.966326497

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.299659829, -2.966326497}
Solution
x = {0.299659829, -2.966326497}