x^4-32x^2+8=0

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Solution for x^4-32x^2+8=0 equation:


Simplifying
x4 + -32x2 + 8 = 0

Reorder the terms:
8 + -32x2 + x4 = 0

Solving
8 + -32x2 + x4 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-8' to each side of the equation.
8 + -32x2 + -8 + x4 = 0 + -8

Reorder the terms:
8 + -8 + -32x2 + x4 = 0 + -8

Combine like terms: 8 + -8 = 0
0 + -32x2 + x4 = 0 + -8
-32x2 + x4 = 0 + -8

Combine like terms: 0 + -8 = -8
-32x2 + x4 = -8

The x term is -32x2.  Take half its coefficient (-16).
Square it (256) and add it to both sides.

Add '256' to each side of the equation.
-32x2 + 256 + x4 = -8 + 256

Reorder the terms:
256 + -32x2 + x4 = -8 + 256

Combine like terms: -8 + 256 = 248
256 + -32x2 + x4 = 248

Factor a perfect square on the left side:
(x2 + -16)(x2 + -16) = 248

Calculate the square root of the right side: 15.748015748

Break this problem into two subproblems by setting 
(x2 + -16) equal to 15.748015748 and -15.748015748.

Subproblem 1

x2 + -16 = 15.748015748 Simplifying x2 + -16 = 15.748015748 Reorder the terms: -16 + x2 = 15.748015748 Solving -16 + x2 = 15.748015748 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '16' to each side of the equation. -16 + 16 + x2 = 15.748015748 + 16 Combine like terms: -16 + 16 = 0 0 + x2 = 15.748015748 + 16 x2 = 15.748015748 + 16 Combine like terms: 15.748015748 + 16 = 31.748015748 x2 = 31.748015748 Simplifying x2 = 31.748015748 Take the square root of each side: x = {-5.634537758, 5.634537758}

Subproblem 2

x2 + -16 = -15.748015748 Simplifying x2 + -16 = -15.748015748 Reorder the terms: -16 + x2 = -15.748015748 Solving -16 + x2 = -15.748015748 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '16' to each side of the equation. -16 + 16 + x2 = -15.748015748 + 16 Combine like terms: -16 + 16 = 0 0 + x2 = -15.748015748 + 16 x2 = -15.748015748 + 16 Combine like terms: -15.748015748 + 16 = 0.251984252 x2 = 0.251984252 Simplifying x2 = 0.251984252 Take the square root of each side: x = {-0.50198033, 0.50198033}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-5.634537758, 5.634537758, -0.50198033, 0.50198033}

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