x(5x+x^2)=x

Solution for x(5x+x^2)=x equation:

Simplifying
x(5x + x2) = x
(5x * x + x2 * x) = x
(5x2 + x3) = x

Solving
5x2 + x3 = x

Solving for variable 'x'.

Reorder the terms:
-1x + 5x2 + x3 = x + -1x

Combine like terms: x + -1x = 0
-1x + 5x2 + x3 = 0

Factor out the Greatest Common Factor (GCF), 'x'.
x(-1 + 5x + x2) = 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 + 5x + x2)' equal to zero and attempt to solve:

Simplifying
-1 + 5x + x2 = 0

Solving
-1 + 5x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + 5x + 1 + x2 = 0 + 1

Reorder the terms:
-1 + 1 + 5x + x2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + 5x + x2 = 0 + 1
5x + x2 = 0 + 1

Combine like terms: 0 + 1 = 1
5x + x2 = 1

The x term is 5x.  Take half its coefficient (2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
5x + 6.25 + x2 = 1 + 6.25

Reorder the terms:
6.25 + 5x + x2 = 1 + 6.25

Combine like terms: 1 + 6.25 = 7.25
6.25 + 5x + x2 = 7.25

Factor a perfect square on the left side:
(x + 2.5)(x + 2.5) = 7.25

Calculate the square root of the right side: 2.692582404

Break this problem into two subproblems by setting 
(x + 2.5) equal to 2.692582404 and -2.692582404.

Subproblem 1
x + 2.5 = 2.692582404

Simplifying
x + 2.5 = 2.692582404

Reorder the terms:
2.5 + x = 2.692582404

Solving
2.5 + x = 2.692582404

Solving for variable 'x'.

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

Add '-2.5' to each side of the equation.
2.5 + -2.5 + x = 2.692582404 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + x = 2.692582404 + -2.5
x = 2.692582404 + -2.5

Combine like terms: 2.692582404 + -2.5 = 0.192582404
x = 0.192582404

Simplifying
x = 0.192582404

Subproblem 2
x + 2.5 = -2.692582404

Simplifying
x + 2.5 = -2.692582404

Reorder the terms:
2.5 + x = -2.692582404

Solving
2.5 + x = -2.692582404

Solving for variable 'x'.

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

Add '-2.5' to each side of the equation.
2.5 + -2.5 + x = -2.692582404 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + x = -2.692582404 + -2.5
x = -2.692582404 + -2.5

Combine like terms: -2.692582404 + -2.5 = -5.192582404
x = -5.192582404

Simplifying
x = -5.192582404

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.192582404, -5.192582404}
Solution
x = {0, 0.192582404, -5.192582404}