6(1-3x^2)-16(1-3x)=0

Solution for 6(1-3x^2)-16(1-3x)=0 equation:

Simplifying
6(1 + -3x2) + -16(1 + -3x) = 0
(1 * 6 + -3x2 * 6) + -16(1 + -3x) = 0
(6 + -18x2) + -16(1 + -3x) = 0
6 + -18x2 + (1 * -16 + -3x * -16) = 0
6 + -18x2 + (-16 + 48x) = 0

Reorder the terms:
6 + -16 + 48x + -18x2 = 0

Combine like terms: 6 + -16 = -10
-10 + 48x + -18x2 = 0

Solving
-10 + 48x + -18x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '2'.
2(-5 + 24x + -9x2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-5 + 24x + -9x2)' equal to zero and attempt to solve:

Simplifying
-5 + 24x + -9x2 = 0

Solving
-5 + 24x + -9x2 = 0

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

Divide each side by '-9'.
0.5555555556 + -2.666666667x + x2 = 0

Move the constant term to the right:

Add '-0.5555555556' to each side of the equation.
0.5555555556 + -2.666666667x + -0.5555555556 + x2 = 0 + -0.5555555556

Reorder the terms:
0.5555555556 + -0.5555555556 + -2.666666667x + x2 = 0 + -0.5555555556

Combine like terms: 0.5555555556 + -0.5555555556 = 0.0000000000
0.0000000000 + -2.666666667x + x2 = 0 + -0.5555555556
-2.666666667x + x2 = 0 + -0.5555555556

Combine like terms: 0 + -0.5555555556 = -0.5555555556
-2.666666667x + x2 = -0.5555555556

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 = -0.5555555556 + 1.777777780

Reorder the terms:
1.777777780 + -2.666666667x + x2 = -0.5555555556 + 1.777777780

Combine like terms: -0.5555555556 + 1.777777780 = 1.2222222244
1.777777780 + -2.666666667x + x2 = 1.2222222244

Factor a perfect square on the left side:
(x + -1.333333334)(x + -1.333333334) = 1.2222222244

Calculate the square root of the right side: 1.105541598

Break this problem into two subproblems by setting 
(x + -1.333333334) equal to 1.105541598 and -1.105541598.

Subproblem 1
x + -1.333333334 = 1.105541598

Simplifying
x + -1.333333334 = 1.105541598

Reorder the terms:
-1.333333334 + x = 1.105541598

Solving
-1.333333334 + x = 1.105541598

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 = 1.105541598 + 1.333333334

Combine like terms: -1.333333334 + 1.333333334 = 0.000000000
0.000000000 + x = 1.105541598 + 1.333333334
x = 1.105541598 + 1.333333334

Combine like terms: 1.105541598 + 1.333333334 = 2.438874932
x = 2.438874932

Simplifying
x = 2.438874932

Subproblem 2
x + -1.333333334 = -1.105541598

Simplifying
x + -1.333333334 = -1.105541598

Reorder the terms:
-1.333333334 + x = -1.105541598

Solving
-1.333333334 + x = -1.105541598

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 = -1.105541598 + 1.333333334

Combine like terms: -1.333333334 + 1.333333334 = 0.000000000
0.000000000 + x = -1.105541598 + 1.333333334
x = -1.105541598 + 1.333333334

Combine like terms: -1.105541598 + 1.333333334 = 0.227791736
x = 0.227791736

Simplifying
x = 0.227791736

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.438874932, 0.227791736}
Solution
x = {2.438874932, 0.227791736}