x^4-8x^2+0.5=0

Solution for x^4-8x^2+0.5=0 equation:

Simplifying
x4 + -8x2 + 0.5 = 0

Reorder the terms:
0.5 + -8x2 + x4 = 0

Solving
0.5 + -8x2 + 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 + -8x2 + -0.5 + x4 = 0 + -0.5

Reorder the terms:
0.5 + -0.5 + -8x2 + x4 = 0 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + -8x2 + x4 = 0 + -0.5
-8x2 + x4 = 0 + -0.5

Combine like terms: 0 + -0.5 = -0.5
-8x2 + x4 = -0.5

The x term is -8x2.  Take half its coefficient (-4).
Square it (16) and add it to both sides.

Add '16' to each side of the equation.
-8x2 + 16 + x4 = -0.5 + 16

Reorder the terms:
16 + -8x2 + x4 = -0.5 + 16

Combine like terms: -0.5 + 16 = 15.5
16 + -8x2 + x4 = 15.5

Factor a perfect square on the left side:
(x2 + -4)(x2 + -4) = 15.5

Calculate the square root of the right side: 3.937003937

Break this problem into two subproblems by setting 
(x2 + -4) equal to 3.937003937 and -3.937003937.

Subproblem 1
x2 + -4 = 3.937003937

Simplifying
x2 + -4 = 3.937003937

Reorder the terms:
-4 + x2 = 3.937003937

Solving
-4 + x2 = 3.937003937

Solving for variable 'x'.

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

Add '4' to each side of the equation.
-4 + 4 + x2 = 3.937003937 + 4

Combine like terms: -4 + 4 = 0
0 + x2 = 3.937003937 + 4
x2 = 3.937003937 + 4

Combine like terms: 3.937003937 + 4 = 7.937003937
x2 = 7.937003937

Simplifying
x2 = 7.937003937

Take the square root of each side:
x = {-2.817268879, 2.817268879}

Subproblem 2
x2 + -4 = -3.937003937

Simplifying
x2 + -4 = -3.937003937

Reorder the terms:
-4 + x2 = -3.937003937

Solving
-4 + x2 = -3.937003937

Solving for variable 'x'.

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

Add '4' to each side of the equation.
-4 + 4 + x2 = -3.937003937 + 4

Combine like terms: -4 + 4 = 0
0 + x2 = -3.937003937 + 4
x2 = -3.937003937 + 4

Combine like terms: -3.937003937 + 4 = 0.062996063
x2 = 0.062996063

Simplifying
x2 = 0.062996063

Take the square root of each side:
x = {-0.250990165, 0.250990165}

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-2.817268879, 2.817268879, -0.250990165, 0.250990165}