5x^4-3x^2+0.25=0

Solution for 5x^4-3x^2+0.25=0 equation:

Simplifying
5x4 + -3x2 + 0.25 = 0

Reorder the terms:
0.25 + -3x2 + 5x4 = 0

Solving
0.25 + -3x2 + 5x4 = 0

Solving for variable 'x'.

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

Divide each side by '5'.
0.05 + -0.6x2 + x4 = 0

Move the constant term to the right:

Add '-0.05' to each side of the equation.
0.05 + -0.6x2 + -0.05 + x4 = 0 + -0.05

Reorder the terms:
0.05 + -0.05 + -0.6x2 + x4 = 0 + -0.05

Combine like terms: 0.05 + -0.05 = 0.00
0.00 + -0.6x2 + x4 = 0 + -0.05
-0.6x2 + x4 = 0 + -0.05

Combine like terms: 0 + -0.05 = -0.05
-0.6x2 + x4 = -0.05

The x term is -0.6x2.  Take half its coefficient (-0.3).
Square it (0.09) and add it to both sides.

Add '0.09' to each side of the equation.
-0.6x2 + 0.09 + x4 = -0.05 + 0.09

Reorder the terms:
0.09 + -0.6x2 + x4 = -0.05 + 0.09

Combine like terms: -0.05 + 0.09 = 0.04
0.09 + -0.6x2 + x4 = 0.04

Factor a perfect square on the left side:
(x2 + -0.3)(x2 + -0.3) = 0.04

Calculate the square root of the right side: 0.2

Break this problem into two subproblems by setting 
(x2 + -0.3) equal to 0.2 and -0.2.

Subproblem 1
x2 + -0.3 = 0.2

Simplifying
x2 + -0.3 = 0.2

Reorder the terms:
-0.3 + x2 = 0.2

Solving
-0.3 + x2 = 0.2

Solving for variable 'x'.

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

Add '0.3' to each side of the equation.
-0.3 + 0.3 + x2 = 0.2 + 0.3

Combine like terms: -0.3 + 0.3 = 0.0
0.0 + x2 = 0.2 + 0.3
x2 = 0.2 + 0.3

Combine like terms: 0.2 + 0.3 = 0.5
x2 = 0.5

Simplifying
x2 = 0.5

Take the square root of each side:
x = {-0.707106781, 0.707106781}

Subproblem 2
x2 + -0.3 = -0.2

Simplifying
x2 + -0.3 = -0.2

Reorder the terms:
-0.3 + x2 = -0.2

Solving
-0.3 + x2 = -0.2

Solving for variable 'x'.

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

Add '0.3' to each side of the equation.
-0.3 + 0.3 + x2 = -0.2 + 0.3

Combine like terms: -0.3 + 0.3 = 0.0
0.0 + x2 = -0.2 + 0.3
x2 = -0.2 + 0.3

Combine like terms: -0.2 + 0.3 = 0.1
x2 = 0.1

Simplifying
x2 = 0.1

Take the square root of each side:
x = {-0.316227766, 0.316227766}

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.707106781, 0.707106781, -0.316227766, 0.316227766}