x^4-12x^2+4=0

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Solution for x^4-12x^2+4=0 equation:


Simplifying
x4 + -12x2 + 4 = 0

Reorder the terms:
4 + -12x2 + x4 = 0

Solving
4 + -12x2 + x4 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-4' to each side of the equation.
4 + -12x2 + -4 + x4 = 0 + -4

Reorder the terms:
4 + -4 + -12x2 + x4 = 0 + -4

Combine like terms: 4 + -4 = 0
0 + -12x2 + x4 = 0 + -4
-12x2 + x4 = 0 + -4

Combine like terms: 0 + -4 = -4
-12x2 + x4 = -4

The x term is -12x2.  Take half its coefficient (-6).
Square it (36) and add it to both sides.

Add '36' to each side of the equation.
-12x2 + 36 + x4 = -4 + 36

Reorder the terms:
36 + -12x2 + x4 = -4 + 36

Combine like terms: -4 + 36 = 32
36 + -12x2 + x4 = 32

Factor a perfect square on the left side:
(x2 + -6)(x2 + -6) = 32

Calculate the square root of the right side: 5.656854249

Break this problem into two subproblems by setting 
(x2 + -6) equal to 5.656854249 and -5.656854249.

Subproblem 1

x2 + -6 = 5.656854249 Simplifying x2 + -6 = 5.656854249 Reorder the terms: -6 + x2 = 5.656854249 Solving -6 + x2 = 5.656854249 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '6' to each side of the equation. -6 + 6 + x2 = 5.656854249 + 6 Combine like terms: -6 + 6 = 0 0 + x2 = 5.656854249 + 6 x2 = 5.656854249 + 6 Combine like terms: 5.656854249 + 6 = 11.656854249 x2 = 11.656854249 Simplifying x2 = 11.656854249 Take the square root of each side: x = {-3.414213562, 3.414213562}

Subproblem 2

x2 + -6 = -5.656854249 Simplifying x2 + -6 = -5.656854249 Reorder the terms: -6 + x2 = -5.656854249 Solving -6 + x2 = -5.656854249 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '6' to each side of the equation. -6 + 6 + x2 = -5.656854249 + 6 Combine like terms: -6 + 6 = 0 0 + x2 = -5.656854249 + 6 x2 = -5.656854249 + 6 Combine like terms: -5.656854249 + 6 = 0.343145751 x2 = 0.343145751 Simplifying x2 = 0.343145751 Take the square root of each side: x = {-0.585786438, 0.585786438}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-3.414213562, 3.414213562, -0.585786438, 0.585786438}

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