x^4-x^2+0.0625=0

Solution for x^4-x^2+0.0625=0 equation:

Simplifying
x4 + -1x2 + 0.0625 = 0

Reorder the terms:
0.0625 + -1x2 + x4 = 0

Solving
0.0625 + -1x2 + x4 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-0.0625' to each side of the equation.
0.0625 + -1x2 + -0.0625 + x4 = 0 + -0.0625

Reorder the terms:
0.0625 + -0.0625 + -1x2 + x4 = 0 + -0.0625

Combine like terms: 0.0625 + -0.0625 = 0.0000
0.0000 + -1x2 + x4 = 0 + -0.0625
-1x2 + x4 = 0 + -0.0625

Combine like terms: 0 + -0.0625 = -0.0625
-1x2 + x4 = -0.0625

The x term is -1x2.  Take half its coefficient (-0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1x2 + 0.25 + x4 = -0.0625 + 0.25

Reorder the terms:
0.25 + -1x2 + x4 = -0.0625 + 0.25

Combine like terms: -0.0625 + 0.25 = 0.1875
0.25 + -1x2 + x4 = 0.1875

Factor a perfect square on the left side:
(x2 + -0.5)(x2 + -0.5) = 0.1875

Calculate the square root of the right side: 0.433012702

Break this problem into two subproblems by setting 
(x2 + -0.5) equal to 0.433012702 and -0.433012702.

Subproblem 1
x2 + -0.5 = 0.433012702

Simplifying
x2 + -0.5 = 0.433012702

Reorder the terms:
-0.5 + x2 = 0.433012702

Solving
-0.5 + x2 = 0.433012702

Solving for variable 'x'.

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

Add '0.5' to each side of the equation.
-0.5 + 0.5 + x2 = 0.433012702 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + x2 = 0.433012702 + 0.5
x2 = 0.433012702 + 0.5

Combine like terms: 0.433012702 + 0.5 = 0.933012702
x2 = 0.933012702

Simplifying
x2 = 0.933012702

Take the square root of each side:
x = {-0.965925826, 0.965925826}

Subproblem 2
x2 + -0.5 = -0.433012702

Simplifying
x2 + -0.5 = -0.433012702

Reorder the terms:
-0.5 + x2 = -0.433012702

Solving
-0.5 + x2 = -0.433012702

Solving for variable 'x'.

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

Add '0.5' to each side of the equation.
-0.5 + 0.5 + x2 = -0.433012702 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + x2 = -0.433012702 + 0.5
x2 = -0.433012702 + 0.5

Combine like terms: -0.433012702 + 0.5 = 0.066987298
x2 = 0.066987298

Simplifying
x2 = 0.066987298

Take the square root of each side:
x = {-0.258819045, 0.258819045}

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.965925826, 0.965925826, -0.258819045, 0.258819045}