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

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


Simplifying
0 = x4 + -8x2 + 1

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

Solving
0 = 1 + -8x2 + x4

Solving for variable 'x'.

Combine like terms: 0 + -1 = -1
-1 + 8x2 + -1x4 = 1 + -8x2 + x4 + -1 + 8x2 + -1x4

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

Combine like terms: 1 + -1 = 0
-1 + 8x2 + -1x4 = 0 + -8x2 + 8x2 + x4 + -1x4
-1 + 8x2 + -1x4 = -8x2 + 8x2 + x4 + -1x4

Combine like terms: -8x2 + 8x2 = 0
-1 + 8x2 + -1x4 = 0 + x4 + -1x4
-1 + 8x2 + -1x4 = x4 + -1x4

Combine like terms: x4 + -1x4 = 0
-1 + 8x2 + -1x4 = 0

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

Divide each side by '-1'.
1 + -8x2 + x4 = 0

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + -8x2 + -1 + x4 = 0 + -1

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

Combine like terms: 1 + -1 = 0
0 + -8x2 + x4 = 0 + -1
-8x2 + x4 = 0 + -1

Combine like terms: 0 + -1 = -1
-8x2 + x4 = -1

The x term is -8x2.  Take half its coefficient (-4).
Square it (16) and add it to both sides.

Add '16' to each side of the equation.
-8x2 + 16 + x4 = -1 + 16

Reorder the terms:
16 + -8x2 + x4 = -1 + 16

Combine like terms: -1 + 16 = 15
16 + -8x2 + x4 = 15

Factor a perfect square on the left side:
(x2 + -4)(x2 + -4) = 15

Calculate the square root of the right side: 3.872983346

Break this problem into two subproblems by setting 
(x2 + -4) equal to 3.872983346 and -3.872983346.

Subproblem 1

x2 + -4 = 3.872983346 Simplifying x2 + -4 = 3.872983346 Reorder the terms: -4 + x2 = 3.872983346 Solving -4 + x2 = 3.872983346 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + x2 = 3.872983346 + 4 Combine like terms: -4 + 4 = 0 0 + x2 = 3.872983346 + 4 x2 = 3.872983346 + 4 Combine like terms: 3.872983346 + 4 = 7.872983346 x2 = 7.872983346 Simplifying x2 = 7.872983346 Take the square root of each side: x = {-2.805883701, 2.805883701}

Subproblem 2

x2 + -4 = -3.872983346 Simplifying x2 + -4 = -3.872983346 Reorder the terms: -4 + x2 = -3.872983346 Solving -4 + x2 = -3.872983346 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + x2 = -3.872983346 + 4 Combine like terms: -4 + 4 = 0 0 + x2 = -3.872983346 + 4 x2 = -3.872983346 + 4 Combine like terms: -3.872983346 + 4 = 0.127016654 x2 = 0.127016654 Simplifying x2 = 0.127016654 Take the square root of each side: x = {-0.356393959, 0.356393959}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-2.805883701, 2.805883701, -0.356393959, 0.356393959}

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