3h(h+1)=3h^3

Solution for 3h(h+1)=3h^3 equation:

Simplifying
3h(h + 1) = 3h3

Reorder the terms:
3h(1 + h) = 3h3
(1 * 3h + h * 3h) = 3h3
(3h + 3h2) = 3h3

Solving
3h + 3h2 = 3h3

Solving for variable 'h'.

Combine like terms: 3h3 + -3h3 = 0
3h + 3h2 + -3h3 = 0

Factor out the Greatest Common Factor (GCF), '3h'.
3h(1 + h + -1h2) = 0

Ignore the factor 3.
Subproblem 1
Set the factor 'h' equal to zero and attempt to solve:

Simplifying
h = 0

Solving
h = 0

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

Simplifying
h = 0
Subproblem 2
Set the factor '(1 + h + -1h2)' equal to zero and attempt to solve:

Simplifying
1 + h + -1h2 = 0

Solving
1 + h + -1h2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-1 + -1h + h2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -1h + 1 + h2 = 0 + 1

Reorder the terms:
-1 + 1 + -1h + h2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -1h + h2 = 0 + 1
-1h + h2 = 0 + 1

Combine like terms: 0 + 1 = 1
-1h + h2 = 1

The h term is h.  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.
-1h + 0.25 + h2 = 1 + 0.25

Reorder the terms:
0.25 + -1h + h2 = 1 + 0.25

Combine like terms: 1 + 0.25 = 1.25
0.25 + -1h + h2 = 1.25

Factor a perfect square on the left side:
(h + 0.5)(h + 0.5) = 1.25

Calculate the square root of the right side: 1.118033989

Break this problem into two subproblems by setting 
(h + 0.5) equal to 1.118033989 and -1.118033989.

Subproblem 1
h + 0.5 = 1.118033989

Simplifying
h + 0.5 = 1.118033989

Reorder the terms:
0.5 + h = 1.118033989

Solving
0.5 + h = 1.118033989

Solving for variable 'h'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + h = 1.118033989 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + h = 1.118033989 + -0.5
h = 1.118033989 + -0.5

Combine like terms: 1.118033989 + -0.5 = 0.618033989
h = 0.618033989

Simplifying
h = 0.618033989

Subproblem 2
h + 0.5 = -1.118033989

Simplifying
h + 0.5 = -1.118033989

Reorder the terms:
0.5 + h = -1.118033989

Solving
0.5 + h = -1.118033989

Solving for variable 'h'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + h = -1.118033989 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + h = -1.118033989 + -0.5
h = -1.118033989 + -0.5

Combine like terms: -1.118033989 + -0.5 = -1.618033989
h = -1.618033989

Simplifying
h = -1.618033989

Solution
The solution to the problem is based on the solutions
from the subproblems.
h = {0.618033989, -1.618033989}
Solution
h = {0, 0.618033989, -1.618033989}