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

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


Simplifying
4x4 + -8x2 + 1 = 0

Reorder the terms:
1 + -8x2 + 4x4 = 0

Solving
1 + -8x2 + 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 + -2x2 + x4 = 0

Move the constant term to the right:

Add '-0.25' to each side of the equation.
0.25 + -2x2 + -0.25 + x4 = 0 + -0.25

Reorder the terms:
0.25 + -0.25 + -2x2 + x4 = 0 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + -2x2 + x4 = 0 + -0.25
-2x2 + x4 = 0 + -0.25

Combine like terms: 0 + -0.25 = -0.25
-2x2 + x4 = -0.25

The x term is -2x2.  Take half its coefficient (-1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
-2x2 + 1 + x4 = -0.25 + 1

Reorder the terms:
1 + -2x2 + x4 = -0.25 + 1

Combine like terms: -0.25 + 1 = 0.75
1 + -2x2 + x4 = 0.75

Factor a perfect square on the left side:
(x2 + -1)(x2 + -1) = 0.75

Calculate the square root of the right side: 0.866025404

Break this problem into two subproblems by setting 
(x2 + -1) equal to 0.866025404 and -0.866025404.

Subproblem 1

x2 + -1 = 0.866025404 Simplifying x2 + -1 = 0.866025404 Reorder the terms: -1 + x2 = 0.866025404 Solving -1 + x2 = 0.866025404 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + x2 = 0.866025404 + 1 Combine like terms: -1 + 1 = 0 0 + x2 = 0.866025404 + 1 x2 = 0.866025404 + 1 Combine like terms: 0.866025404 + 1 = 1.866025404 x2 = 1.866025404 Simplifying x2 = 1.866025404 Take the square root of each side: x = {-1.366025404, 1.366025404}

Subproblem 2

x2 + -1 = -0.866025404 Simplifying x2 + -1 = -0.866025404 Reorder the terms: -1 + x2 = -0.866025404 Solving -1 + x2 = -0.866025404 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + x2 = -0.866025404 + 1 Combine like terms: -1 + 1 = 0 0 + x2 = -0.866025404 + 1 x2 = -0.866025404 + 1 Combine like terms: -0.866025404 + 1 = 0.133974596 x2 = 0.133974596 Simplifying x2 = 0.133974596 Take the square root of each side: x = {-0.366025403, 0.366025403}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-1.366025404, 1.366025404, -0.366025403, 0.366025403}

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