x^4-6x^2+1=0

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Solution for x^4-6x^2+1=0 equation:


Simplifying
x4 + -6x2 + 1 = 0

Reorder the terms:
1 + -6x2 + x4 = 0

Solving
1 + -6x2 + x4 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + -6x2 + -1 + x4 = 0 + -1

Reorder the terms:
1 + -1 + -6x2 + x4 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + -6x2 + x4 = 0 + -1
-6x2 + x4 = 0 + -1

Combine like terms: 0 + -1 = -1
-6x2 + x4 = -1

The x term is -6x2.  Take half its coefficient (-3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
-6x2 + 9 + x4 = -1 + 9

Reorder the terms:
9 + -6x2 + x4 = -1 + 9

Combine like terms: -1 + 9 = 8
9 + -6x2 + x4 = 8

Factor a perfect square on the left side:
(x2 + -3)(x2 + -3) = 8

Calculate the square root of the right side: 2.828427125

Break this problem into two subproblems by setting 
(x2 + -3) equal to 2.828427125 and -2.828427125.

Subproblem 1

x2 + -3 = 2.828427125 Simplifying x2 + -3 = 2.828427125 Reorder the terms: -3 + x2 = 2.828427125 Solving -3 + x2 = 2.828427125 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + x2 = 2.828427125 + 3 Combine like terms: -3 + 3 = 0 0 + x2 = 2.828427125 + 3 x2 = 2.828427125 + 3 Combine like terms: 2.828427125 + 3 = 5.828427125 x2 = 5.828427125 Simplifying x2 = 5.828427125 Take the square root of each side: x = {-2.414213562, 2.414213562}

Subproblem 2

x2 + -3 = -2.828427125 Simplifying x2 + -3 = -2.828427125 Reorder the terms: -3 + x2 = -2.828427125 Solving -3 + x2 = -2.828427125 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + x2 = -2.828427125 + 3 Combine like terms: -3 + 3 = 0 0 + x2 = -2.828427125 + 3 x2 = -2.828427125 + 3 Combine like terms: -2.828427125 + 3 = 0.171572875 x2 = 0.171572875 Simplifying x2 = 0.171572875 Take the square root of each side: x = {-0.414213562, 0.414213562}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-2.414213562, 2.414213562, -0.414213562, 0.414213562}

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