48x^4+16x^2-8=0

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Solution for 48x^4+16x^2-8=0 equation: 


Simplifying
48x4 + 16x2 + -8 = 0

Reorder the terms:
-8 + 16x2 + 48x4 = 0

Solving
-8 + 16x2 + 48x4 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '8'.
8(-1 + 2x2 + 6x4) = 0

Ignore the factor 8.

Subproblem 1
Set the factor '(-1 + 2x2 + 6x4)' equal to zero and attempt to solve:

Simplifying
-1 + 2x2 + 6x4 = 0

Solving
-1 + 2x2 + 6x4 = 0

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

Divide each side by '6'.
-0.1666666667 + 0.3333333333x2 + x4 = 0

Move the constant term to the right:

Add '0.1666666667' to each side of the equation.
-0.1666666667 + 0.3333333333x2 + 0.1666666667 + x4 = 0 + 0.1666666667

Reorder the terms:
-0.1666666667 + 0.1666666667 + 0.3333333333x2 + x4 = 0 + 0.1666666667

Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000
0.0000000000 + 0.3333333333x2 + x4 = 0 + 0.1666666667
0.3333333333x2 + x4 = 0 + 0.1666666667

Combine like terms: 0 + 0.1666666667 = 0.1666666667
0.3333333333x2 + x4 = 0.1666666667

The x term is 0.3333333333x2.  Take half its coefficient (0.1666666667).
Square it (0.02777777779) and add it to both sides.

Add '0.02777777779' to each side of the equation.
0.3333333333x2 + 0.02777777779 + x4 = 0.1666666667 + 0.02777777779

Reorder the terms:
0.02777777779 + 0.3333333333x2 + x4 = 0.1666666667 + 0.02777777779

Combine like terms: 0.1666666667 + 0.02777777779 = 0.19444444449
0.02777777779 + 0.3333333333x2 + x4 = 0.19444444449

Factor a perfect square on the left side:
(x2 + 0.1666666667)(x2 + 0.1666666667) = 0.19444444449

Calculate the square root of the right side: 0.440958552

Break this problem into two subproblems by setting 
(x2 + 0.1666666667) equal to 0.440958552 and -0.440958552.


Subproblem 1
x2 + 0.1666666667 = 0.440958552

Simplifying
x2 + 0.1666666667 = 0.440958552

Reorder the terms:
0.1666666667 + x2 = 0.440958552

Solving
0.1666666667 + x2 = 0.440958552

Solving for variable 'x'.

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

Add '-0.1666666667' to each side of the equation.
0.1666666667 + -0.1666666667 + x2 = 0.440958552 + -0.1666666667

Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000
0.0000000000 + x2 = 0.440958552 + -0.1666666667
x2 = 0.440958552 + -0.1666666667

Combine like terms: 0.440958552 + -0.1666666667 = 0.2742918853
x2 = 0.2742918853

Simplifying
x2 = 0.2742918853

Take the square root of each side:
x = {-0.523728828, 0.523728828}


Subproblem 2
x2 + 0.1666666667 = -0.440958552

Simplifying
x2 + 0.1666666667 = -0.440958552

Reorder the terms:
0.1666666667 + x2 = -0.440958552

Solving
0.1666666667 + x2 = -0.440958552

Solving for variable 'x'.

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

Add '-0.1666666667' to each side of the equation.
0.1666666667 + -0.1666666667 + x2 = -0.440958552 + -0.1666666667

Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000
0.0000000000 + x2 = -0.440958552 + -0.1666666667
x2 = -0.440958552 + -0.1666666667

Combine like terms: -0.440958552 + -0.1666666667 = -0.6076252187
x2 = -0.6076252187

Simplifying
x2 = -0.6076252187

Reorder the terms:
0.6076252187 + x2 = -0.6076252187 + 0.6076252187

Combine like terms: -0.6076252187 + 0.6076252187 = 0.0000000000
0.6076252187 + x2 = 0.0000000000

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.

This subproblem is being ignored because a solution could not be determined.

The solution to this equation could not be determined.

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