3x^4-6x^2+1=0

Solution for 3x^4-6x^2+1=0 equation:

Simplifying
3x4 + -6x2 + 1 = 0

Reorder the terms:
1 + -6x2 + 3x4 = 0

Solving
1 + -6x2 + 3x4 = 0

Solving for variable 'x'.

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

Divide each side by '3'.
0.3333333333 + -2x2 + x4 = 0

Move the constant term to the right:

Add '-0.3333333333' to each side of the equation.
0.3333333333 + -2x2 + -0.3333333333 + x4 = 0 + -0.3333333333

Reorder the terms:
0.3333333333 + -0.3333333333 + -2x2 + x4 = 0 + -0.3333333333

Combine like terms: 0.3333333333 + -0.3333333333 = 0.0000000000
0.0000000000 + -2x2 + x4 = 0 + -0.3333333333
-2x2 + x4 = 0 + -0.3333333333

Combine like terms: 0 + -0.3333333333 = -0.3333333333
-2x2 + x4 = -0.3333333333

The x term is -2x2.  Take half its coefficient (-1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
-2x2 + 1 + x4 = -0.3333333333 + 1

Reorder the terms:
1 + -2x2 + x4 = -0.3333333333 + 1

Combine like terms: -0.3333333333 + 1 = 0.6666666667
1 + -2x2 + x4 = 0.6666666667

Factor a perfect square on the left side:
(x2 + -1)(x2 + -1) = 0.6666666667

Calculate the square root of the right side: 0.816496581

Break this problem into two subproblems by setting 
(x2 + -1) equal to 0.816496581 and -0.816496581.

Subproblem 1
x2 + -1 = 0.816496581

Simplifying
x2 + -1 = 0.816496581

Reorder the terms:
-1 + x2 = 0.816496581

Solving
-1 + x2 = 0.816496581

Solving for variable 'x'.

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

Add '1' to each side of the equation.
-1 + 1 + x2 = 0.816496581 + 1

Combine like terms: -1 + 1 = 0
0 + x2 = 0.816496581 + 1
x2 = 0.816496581 + 1

Combine like terms: 0.816496581 + 1 = 1.816496581
x2 = 1.816496581

Simplifying
x2 = 1.816496581

Take the square root of each side:
x = {-1.347774677, 1.347774677}

Subproblem 2
x2 + -1 = -0.816496581

Simplifying
x2 + -1 = -0.816496581

Reorder the terms:
-1 + x2 = -0.816496581

Solving
-1 + x2 = -0.816496581

Solving for variable 'x'.

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

Add '1' to each side of the equation.
-1 + 1 + x2 = -0.816496581 + 1

Combine like terms: -1 + 1 = 0
0 + x2 = -0.816496581 + 1
x2 = -0.816496581 + 1

Combine like terms: -0.816496581 + 1 = 0.183503419
x2 = 0.183503419

Simplifying
x2 = 0.183503419

Take the square root of each side:
x = {-0.428372991, 0.428372991}

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-1.347774677, 1.347774677, -0.428372991, 0.428372991}