3y^4-6y^3=3y^2

Solution for 3y^4-6y^3=3y^2 equation:

Simplifying
3y4 + -6y3 = 3y2

Reorder the terms:
-6y3 + 3y4 = 3y2

Solving
-6y3 + 3y4 = 3y2

Solving for variable 'y'.

Reorder the terms:
-3y2 + -6y3 + 3y4 = 3y2 + -3y2

Combine like terms: 3y2 + -3y2 = 0
-3y2 + -6y3 + 3y4 = 0

Factor out the Greatest Common Factor (GCF), '3y2'.
3y2(-1 + -2y + y2) = 0

Ignore the factor 3.
Subproblem 1
Set the factor 'y2' equal to zero and attempt to solve:

Simplifying
y2 = 0

Solving
y2 = 0

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

Simplifying
y2 = 0

Take the square root of each side:
y = {0}
Subproblem 2
Set the factor '(-1 + -2y + y2)' equal to zero and attempt to solve:

Simplifying
-1 + -2y + y2 = 0

Solving
-1 + -2y + y2 = 0

Begin completing the square.

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -2y + 1 + y2 = 0 + 1

Reorder the terms:
-1 + 1 + -2y + y2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -2y + y2 = 0 + 1
-2y + y2 = 0 + 1

Combine like terms: 0 + 1 = 1
-2y + y2 = 1

The y term is -2y.  Take half its coefficient (-1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
-2y + 1 + y2 = 1 + 1

Reorder the terms:
1 + -2y + y2 = 1 + 1

Combine like terms: 1 + 1 = 2
1 + -2y + y2 = 2

Factor a perfect square on the left side:
(y + -1)(y + -1) = 2

Calculate the square root of the right side: 1.414213562

Break this problem into two subproblems by setting 
(y + -1) equal to 1.414213562 and -1.414213562.

Subproblem 1
y + -1 = 1.414213562

Simplifying
y + -1 = 1.414213562

Reorder the terms:
-1 + y = 1.414213562

Solving
-1 + y = 1.414213562

Solving for variable 'y'.

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

Add '1' to each side of the equation.
-1 + 1 + y = 1.414213562 + 1

Combine like terms: -1 + 1 = 0
0 + y = 1.414213562 + 1
y = 1.414213562 + 1

Combine like terms: 1.414213562 + 1 = 2.414213562
y = 2.414213562

Simplifying
y = 2.414213562

Subproblem 2
y + -1 = -1.414213562

Simplifying
y + -1 = -1.414213562

Reorder the terms:
-1 + y = -1.414213562

Solving
-1 + y = -1.414213562

Solving for variable 'y'.

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

Add '1' to each side of the equation.
-1 + 1 + y = -1.414213562 + 1

Combine like terms: -1 + 1 = 0
0 + y = -1.414213562 + 1
y = -1.414213562 + 1

Combine like terms: -1.414213562 + 1 = -0.414213562
y = -0.414213562

Simplifying
y = -0.414213562

Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {2.414213562, -0.414213562}
Solution
y = {0, 2.414213562, -0.414213562}