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

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

Simplifying
2x4 + -6x2 + 3 = 0

Reorder the terms:
3 + -6x2 + 2x4 = 0

Solving
3 + -6x2 + 2x4 = 0

Solving for variable 'x'.

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

Divide each side by '2'.
1.5 + -3x2 + x4 = 0

Move the constant term to the right:

Add '-1.5' to each side of the equation.
1.5 + -3x2 + -1.5 + x4 = 0 + -1.5

Reorder the terms:
1.5 + -1.5 + -3x2 + x4 = 0 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + -3x2 + x4 = 0 + -1.5
-3x2 + x4 = 0 + -1.5

Combine like terms: 0 + -1.5 = -1.5
-3x2 + x4 = -1.5

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 = -1.5 + 2.25

Reorder the terms:
2.25 + -3x2 + x4 = -1.5 + 2.25

Combine like terms: -1.5 + 2.25 = 0.75
2.25 + -3x2 + x4 = 0.75

Factor a perfect square on the left side:
(x2 + -1.5)(x2 + -1.5) = 0.75

Calculate the square root of the right side: 0.866025404

Break this problem into two subproblems by setting 
(x2 + -1.5) equal to 0.866025404 and -0.866025404.

Subproblem 1
x2 + -1.5 = 0.866025404

Simplifying
x2 + -1.5 = 0.866025404

Reorder the terms:
-1.5 + x2 = 0.866025404

Solving
-1.5 + x2 = 0.866025404

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 = 0.866025404 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + x2 = 0.866025404 + 1.5
x2 = 0.866025404 + 1.5

Combine like terms: 0.866025404 + 1.5 = 2.366025404
x2 = 2.366025404

Simplifying
x2 = 2.366025404

Take the square root of each side:
x = {-1.538189001, 1.538189001}

Subproblem 2
x2 + -1.5 = -0.866025404

Simplifying
x2 + -1.5 = -0.866025404

Reorder the terms:
-1.5 + x2 = -0.866025404

Solving
-1.5 + x2 = -0.866025404

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 = -0.866025404 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + x2 = -0.866025404 + 1.5
x2 = -0.866025404 + 1.5

Combine like terms: -0.866025404 + 1.5 = 0.633974596
x2 = 0.633974596

Simplifying
x2 = 0.633974596

Take the square root of each side:
x = {-0.796225217, 0.796225217}

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-1.538189001, 1.538189001, -0.796225217, 0.796225217}