3x^4-8x^2+1=0

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Solution for 3x^4-8x^2+1=0 equation:


Simplifying
3x4 + -8x2 + 1 = 0

Reorder the terms:
1 + -8x2 + 3x4 = 0

Solving
1 + -8x2 + 3x4 = 0

Solving for variable 'x'.

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

Divide each side by '3'.
0.3333333333 + -2.666666667x2 + x4 = 0

Move the constant term to the right:

Add '-0.3333333333' to each side of the equation.
0.3333333333 + -2.666666667x2 + -0.3333333333 + x4 = 0 + -0.3333333333

Reorder the terms:
0.3333333333 + -0.3333333333 + -2.666666667x2 + x4 = 0 + -0.3333333333

Combine like terms: 0.3333333333 + -0.3333333333 = 0.0000000000
0.0000000000 + -2.666666667x2 + x4 = 0 + -0.3333333333
-2.666666667x2 + x4 = 0 + -0.3333333333

Combine like terms: 0 + -0.3333333333 = -0.3333333333
-2.666666667x2 + x4 = -0.3333333333

The x term is -2.666666667x2.  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.
-2.666666667x2 + 1.777777780 + x4 = -0.3333333333 + 1.777777780

Reorder the terms:
1.777777780 + -2.666666667x2 + x4 = -0.3333333333 + 1.777777780

Combine like terms: -0.3333333333 + 1.777777780 = 1.4444444467
1.777777780 + -2.666666667x2 + x4 = 1.4444444467

Factor a perfect square on the left side:
(x2 + -1.333333334)(x2 + -1.333333334) = 1.4444444467

Calculate the square root of the right side: 1.201850426

Break this problem into two subproblems by setting 
(x2 + -1.333333334) equal to 1.201850426 and -1.201850426.

Subproblem 1

x2 + -1.333333334 = 1.201850426 Simplifying x2 + -1.333333334 = 1.201850426 Reorder the terms: -1.333333334 + x2 = 1.201850426 Solving -1.333333334 + x2 = 1.201850426 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.333333334' to each side of the equation. -1.333333334 + 1.333333334 + x2 = 1.201850426 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + x2 = 1.201850426 + 1.333333334 x2 = 1.201850426 + 1.333333334 Combine like terms: 1.201850426 + 1.333333334 = 2.53518376 x2 = 2.53518376 Simplifying x2 = 2.53518376 Take the square root of each side: x = {-1.592226039, 1.592226039}

Subproblem 2

x2 + -1.333333334 = -1.201850426 Simplifying x2 + -1.333333334 = -1.201850426 Reorder the terms: -1.333333334 + x2 = -1.201850426 Solving -1.333333334 + x2 = -1.201850426 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.333333334' to each side of the equation. -1.333333334 + 1.333333334 + x2 = -1.201850426 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + x2 = -1.201850426 + 1.333333334 x2 = -1.201850426 + 1.333333334 Combine like terms: -1.201850426 + 1.333333334 = 0.131482908 x2 = 0.131482908 Simplifying x2 = 0.131482908 Take the square root of each side: x = {-0.36260572, 0.36260572}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-1.592226039, 1.592226039, -0.36260572, 0.36260572}

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