4x^4-12x^2+1=0

Solution for 4x^4-12x^2+1=0 equation:

Simplifying
4x4 + -12x2 + 1 = 0

Reorder the terms:
1 + -12x2 + 4x4 = 0

Solving
1 + -12x2 + 4x4 = 0

Solving for variable 'x'.

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

Divide each side by '4'.
0.25 + -3x2 + x4 = 0

Move the constant term to the right:

Add '-0.25' to each side of the equation.
0.25 + -3x2 + -0.25 + x4 = 0 + -0.25

Reorder the terms:
0.25 + -0.25 + -3x2 + x4 = 0 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + -3x2 + x4 = 0 + -0.25
-3x2 + x4 = 0 + -0.25

Combine like terms: 0 + -0.25 = -0.25
-3x2 + x4 = -0.25

The x term is -3x2.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3x2 + 2.25 + x4 = -0.25 + 2.25

Reorder the terms:
2.25 + -3x2 + x4 = -0.25 + 2.25

Combine like terms: -0.25 + 2.25 = 2
2.25 + -3x2 + x4 = 2

Factor a perfect square on the left side:
(x2 + -1.5)(x2 + -1.5) = 2

Calculate the square root of the right side: 1.414213562

Break this problem into two subproblems by setting 
(x2 + -1.5) equal to 1.414213562 and -1.414213562.

Subproblem 1
x2 + -1.5 = 1.414213562

Simplifying
x2 + -1.5 = 1.414213562

Reorder the terms:
-1.5 + x2 = 1.414213562

Solving
-1.5 + x2 = 1.414213562

Solving for variable 'x'.

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

Add '1.5' to each side of the equation.
-1.5 + 1.5 + x2 = 1.414213562 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + x2 = 1.414213562 + 1.5
x2 = 1.414213562 + 1.5

Combine like terms: 1.414213562 + 1.5 = 2.914213562
x2 = 2.914213562

Simplifying
x2 = 2.914213562

Take the square root of each side:
x = {-1.707106781, 1.707106781}

Subproblem 2
x2 + -1.5 = -1.414213562

Simplifying
x2 + -1.5 = -1.414213562

Reorder the terms:
-1.5 + x2 = -1.414213562

Solving
-1.5 + x2 = -1.414213562

Solving for variable 'x'.

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

Add '1.5' to each side of the equation.
-1.5 + 1.5 + x2 = -1.414213562 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + x2 = -1.414213562 + 1.5
x2 = -1.414213562 + 1.5

Combine like terms: -1.414213562 + 1.5 = 0.085786438
x2 = 0.085786438

Simplifying
x2 = 0.085786438

Take the square root of each side:
x = {-0.292893219, 0.292893219}

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-1.707106781, 1.707106781, -0.292893219, 0.292893219}