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

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


Simplifying
x4 + -4x2 + 2 = 0

Reorder the terms:
2 + -4x2 + x4 = 0

Solving
2 + -4x2 + x4 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-2' to each side of the equation.
2 + -4x2 + -2 + x4 = 0 + -2

Reorder the terms:
2 + -2 + -4x2 + x4 = 0 + -2

Combine like terms: 2 + -2 = 0
0 + -4x2 + x4 = 0 + -2
-4x2 + x4 = 0 + -2

Combine like terms: 0 + -2 = -2
-4x2 + x4 = -2

The x term is -4x2.  Take half its coefficient (-2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
-4x2 + 4 + x4 = -2 + 4

Reorder the terms:
4 + -4x2 + x4 = -2 + 4

Combine like terms: -2 + 4 = 2
4 + -4x2 + x4 = 2

Factor a perfect square on the left side:
(x2 + -2)(x2 + -2) = 2

Calculate the square root of the right side: 1.414213562

Break this problem into two subproblems by setting 
(x2 + -2) equal to 1.414213562 and -1.414213562.

Subproblem 1

x2 + -2 = 1.414213562 Simplifying x2 + -2 = 1.414213562 Reorder the terms: -2 + x2 = 1.414213562 Solving -2 + x2 = 1.414213562 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + x2 = 1.414213562 + 2 Combine like terms: -2 + 2 = 0 0 + x2 = 1.414213562 + 2 x2 = 1.414213562 + 2 Combine like terms: 1.414213562 + 2 = 3.414213562 x2 = 3.414213562 Simplifying x2 = 3.414213562 Take the square root of each side: x = {-1.847759065, 1.847759065}

Subproblem 2

x2 + -2 = -1.414213562 Simplifying x2 + -2 = -1.414213562 Reorder the terms: -2 + x2 = -1.414213562 Solving -2 + x2 = -1.414213562 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + x2 = -1.414213562 + 2 Combine like terms: -2 + 2 = 0 0 + x2 = -1.414213562 + 2 x2 = -1.414213562 + 2 Combine like terms: -1.414213562 + 2 = 0.585786438 x2 = 0.585786438 Simplifying x2 = 0.585786438 Take the square root of each side: x = {-0.765366865, 0.765366865}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-1.847759065, 1.847759065, -0.765366865, 0.765366865}

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