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

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


Simplifying
x3 + 3x2 + -6x + 3x = 0

Reorder the terms:
-6x + 3x + 3x2 + x3 = 0

Combine like terms: -6x + 3x = -3x
-3x + 3x2 + x3 = 0

Solving
-3x + 3x2 + x3 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), 'x'.
x(-3 + 3x + 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 '(-3 + 3x + x2)' equal to zero and attempt to solve:

Simplifying
-3 + 3x + x2 = 0

Solving
-3 + 3x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '3' to each side of the equation.
-3 + 3x + 3 + x2 = 0 + 3

Reorder the terms:
-3 + 3 + 3x + x2 = 0 + 3

Combine like terms: -3 + 3 = 0
0 + 3x + x2 = 0 + 3
3x + x2 = 0 + 3

Combine like terms: 0 + 3 = 3
3x + x2 = 3

The x term is 3x.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3x + 2.25 + x2 = 3 + 2.25

Reorder the terms:
2.25 + 3x + x2 = 3 + 2.25

Combine like terms: 3 + 2.25 = 5.25
2.25 + 3x + x2 = 5.25

Factor a perfect square on the left side:
(x + 1.5)(x + 1.5) = 5.25

Calculate the square root of the right side: 2.291287847

Break this problem into two subproblems by setting 
(x + 1.5) equal to 2.291287847 and -2.291287847.


Subproblem 1
x + 1.5 = 2.291287847

Simplifying
x + 1.5 = 2.291287847

Reorder the terms:
1.5 + x = 2.291287847

Solving
1.5 + x = 2.291287847

Solving for variable 'x'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + x = 2.291287847 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + x = 2.291287847 + -1.5
x = 2.291287847 + -1.5

Combine like terms: 2.291287847 + -1.5 = 0.791287847
x = 0.791287847

Simplifying
x = 0.791287847


Subproblem 2
x + 1.5 = -2.291287847

Simplifying
x + 1.5 = -2.291287847

Reorder the terms:
1.5 + x = -2.291287847

Solving
1.5 + x = -2.291287847

Solving for variable 'x'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + x = -2.291287847 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + x = -2.291287847 + -1.5
x = -2.291287847 + -1.5

Combine like terms: -2.291287847 + -1.5 = -3.791287847
x = -3.791287847

Simplifying
x = -3.791287847


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.791287847, -3.791287847}
Solution
x = {0, 0.791287847, -3.791287847}

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