0=3x^3-8x^2+x

Solution for 0=3x^3-8x^2+x equation:

Simplifying
0 = 3x3 + -8x2 + x

Reorder the terms:
0 = x + -8x2 + 3x3

Solving
0 = x + -8x2 + 3x3

Solving for variable 'x'.
Remove the zero:
-1x + 8x2 + -3x3 = x + -8x2 + 3x3 + -1x + 8x2 + -3x3

Reorder the terms:
-1x + 8x2 + -3x3 = x + -1x + -8x2 + 8x2 + 3x3 + -3x3

Combine like terms: x + -1x = 0
-1x + 8x2 + -3x3 = 0 + -8x2 + 8x2 + 3x3 + -3x3
-1x + 8x2 + -3x3 = -8x2 + 8x2 + 3x3 + -3x3

Combine like terms: -8x2 + 8x2 = 0
-1x + 8x2 + -3x3 = 0 + 3x3 + -3x3
-1x + 8x2 + -3x3 = 3x3 + -3x3

Combine like terms: 3x3 + -3x3 = 0
-1x + 8x2 + -3x3 = 0

Factor out the Greatest Common Factor (GCF), 'x'.
x(-1 + 8x + -3x2) = 0

Subproblem 1
Set the factor 'x' equal to zero and attempt to solve:

Simplifying
x = 0

Solving
x = 0

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

Simplifying
x = 0
Subproblem 2
Set the factor '(-1 + 8x + -3x2)' equal to zero and attempt to solve:

Simplifying
-1 + 8x + -3x2 = 0

Solving
-1 + 8x + -3x2 = 0

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

Divide each side by '-3'.
0.3333333333 + -2.666666667x + x2 = 0

Move the constant term to the right:

Add '-0.3333333333' to each side of the equation.
0.3333333333 + -2.666666667x + -0.3333333333 + x2 = 0 + -0.3333333333

Reorder the terms:
0.3333333333 + -0.3333333333 + -2.666666667x + x2 = 0 + -0.3333333333

Combine like terms: 0.3333333333 + -0.3333333333 = 0.0000000000
0.0000000000 + -2.666666667x + x2 = 0 + -0.3333333333
-2.666666667x + x2 = 0 + -0.3333333333

Combine like terms: 0 + -0.3333333333 = -0.3333333333
-2.666666667x + x2 = -0.3333333333

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 = -0.3333333333 + 1.777777780

Reorder the terms:
1.777777780 + -2.666666667x + x2 = -0.3333333333 + 1.777777780

Combine like terms: -0.3333333333 + 1.777777780 = 1.4444444467
1.777777780 + -2.666666667x + x2 = 1.4444444467

Factor a perfect square on the left side:
(x + -1.333333334)(x + -1.333333334) = 1.4444444467

Calculate the square root of the right side: 1.201850426

Break this problem into two subproblems by setting 
(x + -1.333333334) equal to 1.201850426 and -1.201850426.

Subproblem 1
x + -1.333333334 = 1.201850426

Simplifying
x + -1.333333334 = 1.201850426

Reorder the terms:
-1.333333334 + x = 1.201850426

Solving
-1.333333334 + x = 1.201850426

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 = 1.201850426 + 1.333333334

Combine like terms: -1.333333334 + 1.333333334 = 0.000000000
0.000000000 + x = 1.201850426 + 1.333333334
x = 1.201850426 + 1.333333334

Combine like terms: 1.201850426 + 1.333333334 = 2.53518376
x = 2.53518376

Simplifying
x = 2.53518376

Subproblem 2
x + -1.333333334 = -1.201850426

Simplifying
x + -1.333333334 = -1.201850426

Reorder the terms:
-1.333333334 + x = -1.201850426

Solving
-1.333333334 + x = -1.201850426

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 = -1.201850426 + 1.333333334

Combine like terms: -1.333333334 + 1.333333334 = 0.000000000
0.000000000 + x = -1.201850426 + 1.333333334
x = -1.201850426 + 1.333333334

Combine like terms: -1.201850426 + 1.333333334 = 0.131482908
x = 0.131482908

Simplifying
x = 0.131482908

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.53518376, 0.131482908}
Solution
x = {0, 2.53518376, 0.131482908}