4x^4-2x^2-1=0

Solution for 4x^4-2x^2-1=0 equation:

Simplifying
4x4 + -2x2 + -1 = 0

Reorder the terms:
-1 + -2x2 + 4x4 = 0

Solving
-1 + -2x2 + 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'.
-0.25 + -0.5x2 + x4 = 0

Move the constant term to the right:

Add '0.25' to each side of the equation.
-0.25 + -0.5x2 + 0.25 + x4 = 0 + 0.25

Reorder the terms:
-0.25 + 0.25 + -0.5x2 + x4 = 0 + 0.25

Combine like terms: -0.25 + 0.25 = 0.00
0.00 + -0.5x2 + x4 = 0 + 0.25
-0.5x2 + x4 = 0 + 0.25

Combine like terms: 0 + 0.25 = 0.25
-0.5x2 + x4 = 0.25

The x term is -0.5x2.  Take half its coefficient (-0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
-0.5x2 + 0.0625 + x4 = 0.25 + 0.0625

Reorder the terms:
0.0625 + -0.5x2 + x4 = 0.25 + 0.0625

Combine like terms: 0.25 + 0.0625 = 0.3125
0.0625 + -0.5x2 + x4 = 0.3125

Factor a perfect square on the left side:
(x2 + -0.25)(x2 + -0.25) = 0.3125

Calculate the square root of the right side: 0.559016994

Break this problem into two subproblems by setting 
(x2 + -0.25) equal to 0.559016994 and -0.559016994.

Subproblem 1
x2 + -0.25 = 0.559016994

Simplifying
x2 + -0.25 = 0.559016994

Reorder the terms:
-0.25 + x2 = 0.559016994

Solving
-0.25 + x2 = 0.559016994

Solving for variable 'x'.

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

Add '0.25' to each side of the equation.
-0.25 + 0.25 + x2 = 0.559016994 + 0.25

Combine like terms: -0.25 + 0.25 = 0.00
0.00 + x2 = 0.559016994 + 0.25
x2 = 0.559016994 + 0.25

Combine like terms: 0.559016994 + 0.25 = 0.809016994
x2 = 0.809016994

Simplifying
x2 = 0.809016994

Take the square root of each side:
x = {-0.89945372, 0.89945372}

Subproblem 2
x2 + -0.25 = -0.559016994

Simplifying
x2 + -0.25 = -0.559016994

Reorder the terms:
-0.25 + x2 = -0.559016994

Solving
-0.25 + x2 = -0.559016994

Solving for variable 'x'.

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

Add '0.25' to each side of the equation.
-0.25 + 0.25 + x2 = -0.559016994 + 0.25

Combine like terms: -0.25 + 0.25 = 0.00
0.00 + x2 = -0.559016994 + 0.25
x2 = -0.559016994 + 0.25

Combine like terms: -0.559016994 + 0.25 = -0.309016994
x2 = -0.309016994

Simplifying
x2 = -0.309016994

Reorder the terms:
0.309016994 + x2 = -0.309016994 + 0.309016994

Combine like terms: -0.309016994 + 0.309016994 = 0.000000000
0.309016994 + x2 = 0.000000000

The solution to this equation could not be determined.This subproblem is being ignored because a solution could not be determined.

The solution to this equation could not be determined.