4w^5+22w^4+12w^3=0

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Solution for 4w^5+22w^4+12w^3=0 equation: 


Simplifying
4w5 + 22w4 + 12w3 = 0

Reorder the terms:
12w3 + 22w4 + 4w5 = 0

Solving
12w3 + 22w4 + 4w5 = 0

Solving for variable 'w'.

Factor out the Greatest Common Factor (GCF), '2w3'.
2w3(6 + 11w + 2w2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor 'w3' equal to zero and attempt to solve:

Simplifying
w3 = 0

Solving
w3 = 0

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

Simplifying
w3 = 0

The solution to this equation could not be determined.

This subproblem is being ignored because a solution could not be determined.

Subproblem 2
Set the factor '(6 + 11w + 2w2)' equal to zero and attempt to solve:

Simplifying
6 + 11w + 2w2 = 0

Solving
6 + 11w + 2w2 = 0

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

Divide each side by '2'.
3 + 5.5w + w2 = 0

Move the constant term to the right:

Add '-3' to each side of the equation.
3 + 5.5w + -3 + w2 = 0 + -3

Reorder the terms:
3 + -3 + 5.5w + w2 = 0 + -3

Combine like terms: 3 + -3 = 0
0 + 5.5w + w2 = 0 + -3
5.5w + w2 = 0 + -3

Combine like terms: 0 + -3 = -3
5.5w + w2 = -3

The w term is 5.5w.  Take half its coefficient (2.75).
Square it (7.5625) and add it to both sides.

Add '7.5625' to each side of the equation.
5.5w + 7.5625 + w2 = -3 + 7.5625

Reorder the terms:
7.5625 + 5.5w + w2 = -3 + 7.5625

Combine like terms: -3 + 7.5625 = 4.5625
7.5625 + 5.5w + w2 = 4.5625

Factor a perfect square on the left side:
(w + 2.75)(w + 2.75) = 4.5625

Calculate the square root of the right side: 2.136000936

Break this problem into two subproblems by setting 
(w + 2.75) equal to 2.136000936 and -2.136000936.


Subproblem 1
w + 2.75 = 2.136000936

Simplifying
w + 2.75 = 2.136000936

Reorder the terms:
2.75 + w = 2.136000936

Solving
2.75 + w = 2.136000936

Solving for variable 'w'.

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

Add '-2.75' to each side of the equation.
2.75 + -2.75 + w = 2.136000936 + -2.75

Combine like terms: 2.75 + -2.75 = 0.00
0.00 + w = 2.136000936 + -2.75
w = 2.136000936 + -2.75

Combine like terms: 2.136000936 + -2.75 = -0.613999064
w = -0.613999064

Simplifying
w = -0.613999064


Subproblem 2
w + 2.75 = -2.136000936

Simplifying
w + 2.75 = -2.136000936

Reorder the terms:
2.75 + w = -2.136000936

Solving
2.75 + w = -2.136000936

Solving for variable 'w'.

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

Add '-2.75' to each side of the equation.
2.75 + -2.75 + w = -2.136000936 + -2.75

Combine like terms: 2.75 + -2.75 = 0.00
0.00 + w = -2.136000936 + -2.75
w = -2.136000936 + -2.75

Combine like terms: -2.136000936 + -2.75 = -4.886000936
w = -4.886000936

Simplifying
w = -4.886000936


Solution
The solution to the problem is based on the solutions
from the subproblems.
w = {-0.613999064, -4.886000936}
Solution
w = {-0.613999064, -4.886000936}

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