24w^4+64w^2-16=0

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Solution for 24w^4+64w^2-16=0 equation: 


Simplifying
24w4 + 64w2 + -16 = 0

Reorder the terms:
-16 + 64w2 + 24w4 = 0

Solving
-16 + 64w2 + 24w4 = 0

Solving for variable 'w'.

Factor out the Greatest Common Factor (GCF), '8'.
8(-2 + 8w2 + 3w4) = 0

Ignore the factor 8.

Subproblem 1
Set the factor '(-2 + 8w2 + 3w4)' equal to zero and attempt to solve:

Simplifying
-2 + 8w2 + 3w4 = 0

Solving
-2 + 8w2 + 3w4 = 0

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

Divide each side by '3'.
-0.6666666667 + 2.666666667w2 + w4 = 0

Move the constant term to the right:

Add '0.6666666667' to each side of the equation.
-0.6666666667 + 2.666666667w2 + 0.6666666667 + w4 = 0 + 0.6666666667

Reorder the terms:
-0.6666666667 + 0.6666666667 + 2.666666667w2 + w4 = 0 + 0.6666666667

Combine like terms: -0.6666666667 + 0.6666666667 = 0.0000000000
0.0000000000 + 2.666666667w2 + w4 = 0 + 0.6666666667
2.666666667w2 + w4 = 0 + 0.6666666667

Combine like terms: 0 + 0.6666666667 = 0.6666666667
2.666666667w2 + w4 = 0.6666666667

The w term is 2.666666667w2.  Take half its coefficient (1.333333334).
Square it (1.777777780) and add it to both sides.

Add '1.777777780' to each side of the equation.
2.666666667w2 + 1.777777780 + w4 = 0.6666666667 + 1.777777780

Reorder the terms:
1.777777780 + 2.666666667w2 + w4 = 0.6666666667 + 1.777777780

Combine like terms: 0.6666666667 + 1.777777780 = 2.4444444467
1.777777780 + 2.666666667w2 + w4 = 2.4444444467

Factor a perfect square on the left side:
(w2 + 1.333333334)(w2 + 1.333333334) = 2.4444444467

Calculate the square root of the right side: 1.563471921

Break this problem into two subproblems by setting 
(w2 + 1.333333334) equal to 1.563471921 and -1.563471921.


Subproblem 1
w2 + 1.333333334 = 1.563471921

Simplifying
w2 + 1.333333334 = 1.563471921

Reorder the terms:
1.333333334 + w2 = 1.563471921

Solving
1.333333334 + w2 = 1.563471921

Solving for variable 'w'.

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

Add '-1.333333334' to each side of the equation.
1.333333334 + -1.333333334 + w2 = 1.563471921 + -1.333333334

Combine like terms: 1.333333334 + -1.333333334 = 0.000000000
0.000000000 + w2 = 1.563471921 + -1.333333334
w2 = 1.563471921 + -1.333333334

Combine like terms: 1.563471921 + -1.333333334 = 0.230138587
w2 = 0.230138587

Simplifying
w2 = 0.230138587

Take the square root of each side:
w = {-0.479727618, 0.479727618}


Subproblem 2
w2 + 1.333333334 = -1.563471921

Simplifying
w2 + 1.333333334 = -1.563471921

Reorder the terms:
1.333333334 + w2 = -1.563471921

Solving
1.333333334 + w2 = -1.563471921

Solving for variable 'w'.

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

Add '-1.333333334' to each side of the equation.
1.333333334 + -1.333333334 + w2 = -1.563471921 + -1.333333334

Combine like terms: 1.333333334 + -1.333333334 = 0.000000000
0.000000000 + w2 = -1.563471921 + -1.333333334
w2 = -1.563471921 + -1.333333334

Combine like terms: -1.563471921 + -1.333333334 = -2.896805255
w2 = -2.896805255

Simplifying
w2 = -2.896805255

Reorder the terms:
2.896805255 + w2 = -2.896805255 + 2.896805255

Combine like terms: -2.896805255 + 2.896805255 = 0.000000000
2.896805255 + w2 = 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.

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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