10x^3+x^2+x=0

Solution for 10x^3+x^2+x=0 equation:

Simplifying
10x3 + x2 + x = 0

Reorder the terms:
x + x2 + 10x3 = 0

Solving
x + x2 + 10x3 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), 'x'.
x(1 + x + 10x2) = 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 + x + 10x2)' equal to zero and attempt to solve:

Simplifying
1 + x + 10x2 = 0

Solving
1 + x + 10x2 = 0

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

Divide each side by '10'.
0.1 + 0.1x + x2 = 0

Move the constant term to the right:

Add '-0.1' to each side of the equation.
0.1 + 0.1x + -0.1 + x2 = 0 + -0.1

Reorder the terms:
0.1 + -0.1 + 0.1x + x2 = 0 + -0.1

Combine like terms: 0.1 + -0.1 = 0.0
0.0 + 0.1x + x2 = 0 + -0.1
0.1x + x2 = 0 + -0.1

Combine like terms: 0 + -0.1 = -0.1
0.1x + x2 = -0.1

The x term is x.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
0.1x + 0.25 + x2 = -0.1 + 0.25

Reorder the terms:
0.25 + 0.1x + x2 = -0.1 + 0.25

Combine like terms: -0.1 + 0.25 = 0.15
0.25 + 0.1x + x2 = 0.15

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 0.15

Calculate the square root of the right side: 0.387298335

Break this problem into two subproblems by setting 
(x + 0.5) equal to 0.387298335 and -0.387298335.

Subproblem 1
x + 0.5 = 0.387298335

Simplifying
x + 0.5 = 0.387298335

Reorder the terms:
0.5 + x = 0.387298335

Solving
0.5 + x = 0.387298335

Solving for variable 'x'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = 0.387298335 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 0.387298335 + -0.5
x = 0.387298335 + -0.5

Combine like terms: 0.387298335 + -0.5 = -0.112701665
x = -0.112701665

Simplifying
x = -0.112701665

Subproblem 2
x + 0.5 = -0.387298335

Simplifying
x + 0.5 = -0.387298335

Reorder the terms:
0.5 + x = -0.387298335

Solving
0.5 + x = -0.387298335

Solving for variable 'x'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = -0.387298335 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -0.387298335 + -0.5
x = -0.387298335 + -0.5

Combine like terms: -0.387298335 + -0.5 = -0.887298335
x = -0.887298335

Simplifying
x = -0.887298335

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.112701665, -0.887298335}
Solution
x = {0, -0.112701665, -0.887298335}