4x^3-15x^2+6x=0

Solution for 4x^3-15x^2+6x=0 equation:

Simplifying
4x3 + -15x2 + 6x = 0

Reorder the terms:
6x + -15x2 + 4x3 = 0

Solving
6x + -15x2 + 4x3 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), 'x'.
x(6 + -15x + 4x2) = 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 '(6 + -15x + 4x2)' equal to zero and attempt to solve:

Simplifying
6 + -15x + 4x2 = 0

Solving
6 + -15x + 4x2 = 0

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

Divide each side by '4'.
1.5 + -3.75x + x2 = 0

Move the constant term to the right:

Add '-1.5' to each side of the equation.
1.5 + -3.75x + -1.5 + x2 = 0 + -1.5

Reorder the terms:
1.5 + -1.5 + -3.75x + x2 = 0 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + -3.75x + x2 = 0 + -1.5
-3.75x + x2 = 0 + -1.5

Combine like terms: 0 + -1.5 = -1.5
-3.75x + x2 = -1.5

The x term is -3.75x.  Take half its coefficient (-1.875).
Square it (3.515625) and add it to both sides.

Add '3.515625' to each side of the equation.
-3.75x + 3.515625 + x2 = -1.5 + 3.515625

Reorder the terms:
3.515625 + -3.75x + x2 = -1.5 + 3.515625

Combine like terms: -1.5 + 3.515625 = 2.015625
3.515625 + -3.75x + x2 = 2.015625

Factor a perfect square on the left side:
(x + -1.875)(x + -1.875) = 2.015625

Calculate the square root of the right side: 1.419727086

Break this problem into two subproblems by setting 
(x + -1.875) equal to 1.419727086 and -1.419727086.

Subproblem 1
x + -1.875 = 1.419727086

Simplifying
x + -1.875 = 1.419727086

Reorder the terms:
-1.875 + x = 1.419727086

Solving
-1.875 + x = 1.419727086

Solving for variable 'x'.

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

Add '1.875' to each side of the equation.
-1.875 + 1.875 + x = 1.419727086 + 1.875

Combine like terms: -1.875 + 1.875 = 0.000
0.000 + x = 1.419727086 + 1.875
x = 1.419727086 + 1.875

Combine like terms: 1.419727086 + 1.875 = 3.294727086
x = 3.294727086

Simplifying
x = 3.294727086

Subproblem 2
x + -1.875 = -1.419727086

Simplifying
x + -1.875 = -1.419727086

Reorder the terms:
-1.875 + x = -1.419727086

Solving
-1.875 + x = -1.419727086

Solving for variable 'x'.

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

Add '1.875' to each side of the equation.
-1.875 + 1.875 + x = -1.419727086 + 1.875

Combine like terms: -1.875 + 1.875 = 0.000
0.000 + x = -1.419727086 + 1.875
x = -1.419727086 + 1.875

Combine like terms: -1.419727086 + 1.875 = 0.455272914
x = 0.455272914

Simplifying
x = 0.455272914

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.294727086, 0.455272914}
Solution
x = {0, 3.294727086, 0.455272914}