x^4-2x^2+0.5=0

Solution for x^4-2x^2+0.5=0 equation:

Simplifying
x4 + -2x2 + 0.5 = 0

Reorder the terms:
0.5 + -2x2 + x4 = 0

Solving
0.5 + -2x2 + x4 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-0.5' to each side of the equation.
0.5 + -2x2 + -0.5 + x4 = 0 + -0.5

Reorder the terms:
0.5 + -0.5 + -2x2 + x4 = 0 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + -2x2 + x4 = 0 + -0.5
-2x2 + x4 = 0 + -0.5

Combine like terms: 0 + -0.5 = -0.5
-2x2 + x4 = -0.5

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 = -0.5 + 1

Reorder the terms:
1 + -2x2 + x4 = -0.5 + 1

Combine like terms: -0.5 + 1 = 0.5
1 + -2x2 + x4 = 0.5

Factor a perfect square on the left side:
(x2 + -1)(x2 + -1) = 0.5

Calculate the square root of the right side: 0.707106781

Break this problem into two subproblems by setting 
(x2 + -1) equal to 0.707106781 and -0.707106781.

Subproblem 1
x2 + -1 = 0.707106781

Simplifying
x2 + -1 = 0.707106781

Reorder the terms:
-1 + x2 = 0.707106781

Solving
-1 + x2 = 0.707106781

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 = 0.707106781 + 1

Combine like terms: -1 + 1 = 0
0 + x2 = 0.707106781 + 1
x2 = 0.707106781 + 1

Combine like terms: 0.707106781 + 1 = 1.707106781
x2 = 1.707106781

Simplifying
x2 = 1.707106781

Take the square root of each side:
x = {-1.306562965, 1.306562965}

Subproblem 2
x2 + -1 = -0.707106781

Simplifying
x2 + -1 = -0.707106781

Reorder the terms:
-1 + x2 = -0.707106781

Solving
-1 + x2 = -0.707106781

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 = -0.707106781 + 1

Combine like terms: -1 + 1 = 0
0 + x2 = -0.707106781 + 1
x2 = -0.707106781 + 1

Combine like terms: -0.707106781 + 1 = 0.292893219
x2 = 0.292893219

Simplifying
x2 = 0.292893219

Take the square root of each side:
x = {-0.5411961, 0.5411961}

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-1.306562965, 1.306562965, -0.5411961, 0.5411961}