4x^4-10x^2+3=0

Solution for 4x^4-10x^2+3=0 equation:

Simplifying
4x4 + -10x2 + 3 = 0

Reorder the terms:
3 + -10x2 + 4x4 = 0

Solving
3 + -10x2 + 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'.
0.75 + -2.5x2 + x4 = 0

Move the constant term to the right:

Add '-0.75' to each side of the equation.
0.75 + -2.5x2 + -0.75 + x4 = 0 + -0.75

Reorder the terms:
0.75 + -0.75 + -2.5x2 + x4 = 0 + -0.75

Combine like terms: 0.75 + -0.75 = 0.00
0.00 + -2.5x2 + x4 = 0 + -0.75
-2.5x2 + x4 = 0 + -0.75

Combine like terms: 0 + -0.75 = -0.75
-2.5x2 + x4 = -0.75

The x term is -2.5x2.  Take half its coefficient (-1.25).
Square it (1.5625) and add it to both sides.

Add '1.5625' to each side of the equation.
-2.5x2 + 1.5625 + x4 = -0.75 + 1.5625

Reorder the terms:
1.5625 + -2.5x2 + x4 = -0.75 + 1.5625

Combine like terms: -0.75 + 1.5625 = 0.8125
1.5625 + -2.5x2 + x4 = 0.8125

Factor a perfect square on the left side:
(x2 + -1.25)(x2 + -1.25) = 0.8125

Calculate the square root of the right side: 0.901387819

Break this problem into two subproblems by setting 
(x2 + -1.25) equal to 0.901387819 and -0.901387819.

Subproblem 1
x2 + -1.25 = 0.901387819

Simplifying
x2 + -1.25 = 0.901387819

Reorder the terms:
-1.25 + x2 = 0.901387819

Solving
-1.25 + x2 = 0.901387819

Solving for variable 'x'.

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

Add '1.25' to each side of the equation.
-1.25 + 1.25 + x2 = 0.901387819 + 1.25

Combine like terms: -1.25 + 1.25 = 0.00
0.00 + x2 = 0.901387819 + 1.25
x2 = 0.901387819 + 1.25

Combine like terms: 0.901387819 + 1.25 = 2.151387819
x2 = 2.151387819

Simplifying
x2 = 2.151387819

Take the square root of each side:
x = {-1.466760996, 1.466760996}

Subproblem 2
x2 + -1.25 = -0.901387819

Simplifying
x2 + -1.25 = -0.901387819

Reorder the terms:
-1.25 + x2 = -0.901387819

Solving
-1.25 + x2 = -0.901387819

Solving for variable 'x'.

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

Add '1.25' to each side of the equation.
-1.25 + 1.25 + x2 = -0.901387819 + 1.25

Combine like terms: -1.25 + 1.25 = 0.00
0.00 + x2 = -0.901387819 + 1.25
x2 = -0.901387819 + 1.25

Combine like terms: -0.901387819 + 1.25 = 0.348612181
x2 = 0.348612181

Simplifying
x2 = 0.348612181

Take the square root of each side:
x = {-0.590433892, 0.590433892}

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-1.466760996, 1.466760996, -0.590433892, 0.590433892}