0=3x^3+10x^2+4x

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Solution for 0=3x^3+10x^2+4x equation: 


Simplifying
0 = 3x3 + 10x2 + 4x

Reorder the terms:
0 = 4x + 10x2 + 3x3

Solving
0 = 4x + 10x2 + 3x3

Solving for variable 'x'.
Remove the zero:
-4x + -10x2 + -3x3 = 4x + 10x2 + 3x3 + -4x + -10x2 + -3x3

Reorder the terms:
-4x + -10x2 + -3x3 = 4x + -4x + 10x2 + -10x2 + 3x3 + -3x3

Combine like terms: 4x + -4x = 0
-4x + -10x2 + -3x3 = 0 + 10x2 + -10x2 + 3x3 + -3x3
-4x + -10x2 + -3x3 = 10x2 + -10x2 + 3x3 + -3x3

Combine like terms: 10x2 + -10x2 = 0
-4x + -10x2 + -3x3 = 0 + 3x3 + -3x3
-4x + -10x2 + -3x3 = 3x3 + -3x3

Combine like terms: 3x3 + -3x3 = 0
-4x + -10x2 + -3x3 = 0

Factor out the Greatest Common Factor (GCF), '-1x'.
-1x(4 + 10x + 3x2) = 0

Ignore the factor -1.

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 '(4 + 10x + 3x2)' equal to zero and attempt to solve:

Simplifying
4 + 10x + 3x2 = 0

Solving
4 + 10x + 3x2 = 0

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

Divide each side by '3'.
1.333333333 + 3.333333333x + x2 = 0

Move the constant term to the right:

Add '-1.333333333' to each side of the equation.
1.333333333 + 3.333333333x + -1.333333333 + x2 = 0 + -1.333333333

Reorder the terms:
1.333333333 + -1.333333333 + 3.333333333x + x2 = 0 + -1.333333333

Combine like terms: 1.333333333 + -1.333333333 = 0.000000000
0.000000000 + 3.333333333x + x2 = 0 + -1.333333333
3.333333333x + x2 = 0 + -1.333333333

Combine like terms: 0 + -1.333333333 = -1.333333333
3.333333333x + x2 = -1.333333333

The x term is 3.333333333x.  Take half its coefficient (1.666666667).
Square it (2.777777779) and add it to both sides.

Add '2.777777779' to each side of the equation.
3.333333333x + 2.777777779 + x2 = -1.333333333 + 2.777777779

Reorder the terms:
2.777777779 + 3.333333333x + x2 = -1.333333333 + 2.777777779

Combine like terms: -1.333333333 + 2.777777779 = 1.444444446
2.777777779 + 3.333333333x + x2 = 1.444444446

Factor a perfect square on the left side:
(x + 1.666666667)(x + 1.666666667) = 1.444444446

Calculate the square root of the right side: 1.201850426

Break this problem into two subproblems by setting 
(x + 1.666666667) equal to 1.201850426 and -1.201850426.


Subproblem 1
x + 1.666666667 = 1.201850426

Simplifying
x + 1.666666667 = 1.201850426

Reorder the terms:
1.666666667 + x = 1.201850426

Solving
1.666666667 + x = 1.201850426

Solving for variable 'x'.

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

Add '-1.666666667' to each side of the equation.
1.666666667 + -1.666666667 + x = 1.201850426 + -1.666666667

Combine like terms: 1.666666667 + -1.666666667 = 0.000000000
0.000000000 + x = 1.201850426 + -1.666666667
x = 1.201850426 + -1.666666667

Combine like terms: 1.201850426 + -1.666666667 = -0.464816241
x = -0.464816241

Simplifying
x = -0.464816241


Subproblem 2
x + 1.666666667 = -1.201850426

Simplifying
x + 1.666666667 = -1.201850426

Reorder the terms:
1.666666667 + x = -1.201850426

Solving
1.666666667 + x = -1.201850426

Solving for variable 'x'.

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

Add '-1.666666667' to each side of the equation.
1.666666667 + -1.666666667 + x = -1.201850426 + -1.666666667

Combine like terms: 1.666666667 + -1.666666667 = 0.000000000
0.000000000 + x = -1.201850426 + -1.666666667
x = -1.201850426 + -1.666666667

Combine like terms: -1.201850426 + -1.666666667 = -2.868517093
x = -2.868517093

Simplifying
x = -2.868517093


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.464816241, -2.868517093}
Solution
x = {0, -0.464816241, -2.868517093}

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