24x^4+32x^2-128=0

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Solution for 24x^4+32x^2-128=0 equation: 


Simplifying
24x4 + 32x2 + -128 = 0

Reorder the terms:
-128 + 32x2 + 24x4 = 0

Solving
-128 + 32x2 + 24x4 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '8'.
8(-16 + 4x2 + 3x4) = 0

Ignore the factor 8.

Subproblem 1
Set the factor '(-16 + 4x2 + 3x4)' equal to zero and attempt to solve:

Simplifying
-16 + 4x2 + 3x4 = 0

Solving
-16 + 4x2 + 3x4 = 0

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

Divide each side by '3'.
-5.333333333 + 1.333333333x2 + x4 = 0

Move the constant term to the right:

Add '5.333333333' to each side of the equation.
-5.333333333 + 1.333333333x2 + 5.333333333 + x4 = 0 + 5.333333333

Reorder the terms:
-5.333333333 + 5.333333333 + 1.333333333x2 + x4 = 0 + 5.333333333

Combine like terms: -5.333333333 + 5.333333333 = 0.000000000
0.000000000 + 1.333333333x2 + x4 = 0 + 5.333333333
1.333333333x2 + x4 = 0 + 5.333333333

Combine like terms: 0 + 5.333333333 = 5.333333333
1.333333333x2 + x4 = 5.333333333

The x term is 1.333333333x2.  Take half its coefficient (0.6666666665).
Square it (0.4444444442) and add it to both sides.

Add '0.4444444442' to each side of the equation.
1.333333333x2 + 0.4444444442 + x4 = 5.333333333 + 0.4444444442

Reorder the terms:
0.4444444442 + 1.333333333x2 + x4 = 5.333333333 + 0.4444444442

Combine like terms: 5.333333333 + 0.4444444442 = 5.7777777772
0.4444444442 + 1.333333333x2 + x4 = 5.7777777772

Factor a perfect square on the left side:
(x2 + 0.6666666665)(x2 + 0.6666666665) = 5.7777777772

Calculate the square root of the right side: 2.40370085

Break this problem into two subproblems by setting 
(x2 + 0.6666666665) equal to 2.40370085 and -2.40370085.


Subproblem 1
x2 + 0.6666666665 = 2.40370085

Simplifying
x2 + 0.6666666665 = 2.40370085

Reorder the terms:
0.6666666665 + x2 = 2.40370085

Solving
0.6666666665 + x2 = 2.40370085

Solving for variable 'x'.

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

Add '-0.6666666665' to each side of the equation.
0.6666666665 + -0.6666666665 + x2 = 2.40370085 + -0.6666666665

Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000
0.0000000000 + x2 = 2.40370085 + -0.6666666665
x2 = 2.40370085 + -0.6666666665

Combine like terms: 2.40370085 + -0.6666666665 = 1.7370341835
x2 = 1.7370341835

Simplifying
x2 = 1.7370341835

Take the square root of each side:
x = {-1.317965927, 1.317965927}


Subproblem 2
x2 + 0.6666666665 = -2.40370085

Simplifying
x2 + 0.6666666665 = -2.40370085

Reorder the terms:
0.6666666665 + x2 = -2.40370085

Solving
0.6666666665 + x2 = -2.40370085

Solving for variable 'x'.

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

Add '-0.6666666665' to each side of the equation.
0.6666666665 + -0.6666666665 + x2 = -2.40370085 + -0.6666666665

Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000
0.0000000000 + x2 = -2.40370085 + -0.6666666665
x2 = -2.40370085 + -0.6666666665

Combine like terms: -2.40370085 + -0.6666666665 = -3.0703675165
x2 = -3.0703675165

Simplifying
x2 = -3.0703675165

Reorder the terms:
3.0703675165 + x2 = -3.0703675165 + 3.0703675165

Combine like terms: -3.0703675165 + 3.0703675165 = 0.0000000000
3.0703675165 + 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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