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

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


Simplifying
0 = x4 + -4x2 + 1

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

Solving
0 = 1 + -4x2 + x4

Solving for variable 'x'.

Combine like terms: 0 + -1 = -1
-1 + 4x2 + -1x4 = 1 + -4x2 + x4 + -1 + 4x2 + -1x4

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

Combine like terms: 1 + -1 = 0
-1 + 4x2 + -1x4 = 0 + -4x2 + 4x2 + x4 + -1x4
-1 + 4x2 + -1x4 = -4x2 + 4x2 + x4 + -1x4

Combine like terms: -4x2 + 4x2 = 0
-1 + 4x2 + -1x4 = 0 + x4 + -1x4
-1 + 4x2 + -1x4 = x4 + -1x4

Combine like terms: x4 + -1x4 = 0
-1 + 4x2 + -1x4 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
1 + -4x2 + x4 = 0

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + -4x2 + -1 + x4 = 0 + -1

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

Combine like terms: 1 + -1 = 0
0 + -4x2 + x4 = 0 + -1
-4x2 + x4 = 0 + -1

Combine like terms: 0 + -1 = -1
-4x2 + x4 = -1

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 = -1 + 4

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

Combine like terms: -1 + 4 = 3
4 + -4x2 + x4 = 3

Factor a perfect square on the left side:
(x2 + -2)(x2 + -2) = 3

Calculate the square root of the right side: 1.732050808

Break this problem into two subproblems by setting 
(x2 + -2) equal to 1.732050808 and -1.732050808.

Subproblem 1

x2 + -2 = 1.732050808 Simplifying x2 + -2 = 1.732050808 Reorder the terms: -2 + x2 = 1.732050808 Solving -2 + x2 = 1.732050808 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 = 1.732050808 + 2 Combine like terms: -2 + 2 = 0 0 + x2 = 1.732050808 + 2 x2 = 1.732050808 + 2 Combine like terms: 1.732050808 + 2 = 3.732050808 x2 = 3.732050808 Simplifying x2 = 3.732050808 Take the square root of each side: x = {-1.931851653, 1.931851653}

Subproblem 2

x2 + -2 = -1.732050808 Simplifying x2 + -2 = -1.732050808 Reorder the terms: -2 + x2 = -1.732050808 Solving -2 + x2 = -1.732050808 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 = -1.732050808 + 2 Combine like terms: -2 + 2 = 0 0 + x2 = -1.732050808 + 2 x2 = -1.732050808 + 2 Combine like terms: -1.732050808 + 2 = 0.267949192 x2 = 0.267949192 Simplifying x2 = 0.267949192 Take the square root of each side: x = {-0.51763809, 0.51763809}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-1.931851653, 1.931851653, -0.51763809, 0.51763809}

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