(.2+x)x=x^3

Solution for (.2+x)x=x^3 equation:

Simplifying
(0.2 + x) * x = x3

Reorder the terms for easier multiplication:
x(0.2 + x) = x3
(0.2 * x + x * x) = x3
(0.2x + x2) = x3

Solving
0.2x + x2 = x3

Solving for variable 'x'.

Combine like terms: x3 + -1x3 = 0
0.2x + x2 + -1x3 = 0

Factor out the Greatest Common Factor (GCF), 'x'.
x(0.2 + x + -1x2) = 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 '(0.2 + x + -1x2)' equal to zero and attempt to solve:

Simplifying
0.2 + x + -1x2 = 0

Solving
0.2 + x + -1x2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-0.2 + -1x + x2 = 0

Move the constant term to the right:

Add '0.2' to each side of the equation.
-0.2 + -1x + 0.2 + x2 = 0 + 0.2

Reorder the terms:
-0.2 + 0.2 + -1x + x2 = 0 + 0.2

Combine like terms: -0.2 + 0.2 = 0.0
0.0 + -1x + x2 = 0 + 0.2
-1x + x2 = 0 + 0.2

Combine like terms: 0 + 0.2 = 0.2
-1x + x2 = 0.2

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.
-1x + 0.25 + x2 = 0.2 + 0.25

Reorder the terms:
0.25 + -1x + x2 = 0.2 + 0.25

Combine like terms: 0.2 + 0.25 = 0.45
0.25 + -1x + x2 = 0.45

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 0.45

Calculate the square root of the right side: 0.670820393

Break this problem into two subproblems by setting 
(x + 0.5) equal to 0.670820393 and -0.670820393.

Subproblem 1
x + 0.5 = 0.670820393

Simplifying
x + 0.5 = 0.670820393

Reorder the terms:
0.5 + x = 0.670820393

Solving
0.5 + x = 0.670820393

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 = 0.670820393 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 0.670820393 + -0.5
x = 0.670820393 + -0.5

Combine like terms: 0.670820393 + -0.5 = 0.170820393
x = 0.170820393

Simplifying
x = 0.170820393

Subproblem 2
x + 0.5 = -0.670820393

Simplifying
x + 0.5 = -0.670820393

Reorder the terms:
0.5 + x = -0.670820393

Solving
0.5 + x = -0.670820393

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 = -0.670820393 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -0.670820393 + -0.5
x = -0.670820393 + -0.5

Combine like terms: -0.670820393 + -0.5 = -1.170820393
x = -1.170820393

Simplifying
x = -1.170820393

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.170820393, -1.170820393}
Solution
x = {0, 0.170820393, -1.170820393}