2y^4-14y^2+2=0

Solution for 2y^4-14y^2+2=0 equation:

Simplifying
2y4 + -14y2 + 2 = 0

Reorder the terms:
2 + -14y2 + 2y4 = 0

Solving
2 + -14y2 + 2y4 = 0

Solving for variable 'y'.

Factor out the Greatest Common Factor (GCF), '2'.
2(1 + -7y2 + y4) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(1 + -7y2 + y4)' equal to zero and attempt to solve:

Simplifying
1 + -7y2 + y4 = 0

Solving
1 + -7y2 + y4 = 0

Begin completing the square.

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + -7y2 + -1 + y4 = 0 + -1

Reorder the terms:
1 + -1 + -7y2 + y4 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + -7y2 + y4 = 0 + -1
-7y2 + y4 = 0 + -1

Combine like terms: 0 + -1 = -1
-7y2 + y4 = -1

The y term is -7y2.  Take half its coefficient (-3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
-7y2 + 12.25 + y4 = -1 + 12.25

Reorder the terms:
12.25 + -7y2 + y4 = -1 + 12.25

Combine like terms: -1 + 12.25 = 11.25
12.25 + -7y2 + y4 = 11.25

Factor a perfect square on the left side:
(y2 + -3.5)(y2 + -3.5) = 11.25

Calculate the square root of the right side: 3.354101966

Break this problem into two subproblems by setting 
(y2 + -3.5) equal to 3.354101966 and -3.354101966.

Subproblem 1
y2 + -3.5 = 3.354101966

Simplifying
y2 + -3.5 = 3.354101966

Reorder the terms:
-3.5 + y2 = 3.354101966

Solving
-3.5 + y2 = 3.354101966

Solving for variable 'y'.

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

Add '3.5' to each side of the equation.
-3.5 + 3.5 + y2 = 3.354101966 + 3.5

Combine like terms: -3.5 + 3.5 = 0.0
0.0 + y2 = 3.354101966 + 3.5
y2 = 3.354101966 + 3.5

Combine like terms: 3.354101966 + 3.5 = 6.854101966
y2 = 6.854101966

Simplifying
y2 = 6.854101966

Take the square root of each side:
y = {-2.618033989, 2.618033989}

Subproblem 2
y2 + -3.5 = -3.354101966

Simplifying
y2 + -3.5 = -3.354101966

Reorder the terms:
-3.5 + y2 = -3.354101966

Solving
-3.5 + y2 = -3.354101966

Solving for variable 'y'.

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

Add '3.5' to each side of the equation.
-3.5 + 3.5 + y2 = -3.354101966 + 3.5

Combine like terms: -3.5 + 3.5 = 0.0
0.0 + y2 = -3.354101966 + 3.5
y2 = -3.354101966 + 3.5

Combine like terms: -3.354101966 + 3.5 = 0.145898034
y2 = 0.145898034

Simplifying
y2 = 0.145898034

Take the square root of each side:
y = {-0.381966012, 0.381966012}

Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {-2.618033989, 2.618033989, -0.381966012, 0.381966012}
Solution
y = {-2.618033989, 2.618033989, -0.381966012, 0.381966012}