x^3+x^2-x=4x

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


Simplifying
x3 + x2 + -1x = 4x

Reorder the terms:
-1x + x2 + x3 = 4x

Solving
-1x + x2 + x3 = 4x

Solving for variable 'x'.

Reorder the terms:
-1x + -4x + x2 + x3 = 4x + -4x

Combine like terms: -1x + -4x = -5x
-5x + x2 + x3 = 4x + -4x

Combine like terms: 4x + -4x = 0
-5x + x2 + x3 = 0

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

Simplifying
-5 + x + x2 = 0

Solving
-5 + x + x2 = 0

Begin completing the square.

Move the constant term to the right:

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

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

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

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

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.
x + 0.25 + x2 = 5 + 0.25

Reorder the terms:
0.25 + x + x2 = 5 + 0.25

Combine like terms: 5 + 0.25 = 5.25
0.25 + x + x2 = 5.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 5.25

Calculate the square root of the right side: 2.291287847

Break this problem into two subproblems by setting 
(x + 0.5) equal to 2.291287847 and -2.291287847.


Subproblem 1
x + 0.5 = 2.291287847

Simplifying
x + 0.5 = 2.291287847

Reorder the terms:
0.5 + x = 2.291287847

Solving
0.5 + x = 2.291287847

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 = 2.291287847 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 2.291287847 + -0.5
x = 2.291287847 + -0.5

Combine like terms: 2.291287847 + -0.5 = 1.791287847
x = 1.791287847

Simplifying
x = 1.791287847


Subproblem 2
x + 0.5 = -2.291287847

Simplifying
x + 0.5 = -2.291287847

Reorder the terms:
0.5 + x = -2.291287847

Solving
0.5 + x = -2.291287847

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 = -2.291287847 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -2.291287847 + -0.5
x = -2.291287847 + -0.5

Combine like terms: -2.291287847 + -0.5 = -2.791287847
x = -2.791287847

Simplifying
x = -2.791287847


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.791287847, -2.791287847}
Solution
x = {0, 1.791287847, -2.791287847}

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