3x^4-12x^2+1=0

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Solution for 3x^4-12x^2+1=0 equation:


Simplifying
3x4 + -12x2 + 1 = 0

Reorder the terms:
1 + -12x2 + 3x4 = 0

Solving
1 + -12x2 + 3x4 = 0

Solving for variable 'x'.

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

Divide each side by '3'.
0.3333333333 + -4x2 + x4 = 0

Move the constant term to the right:

Add '-0.3333333333' to each side of the equation.
0.3333333333 + -4x2 + -0.3333333333 + x4 = 0 + -0.3333333333

Reorder the terms:
0.3333333333 + -0.3333333333 + -4x2 + x4 = 0 + -0.3333333333

Combine like terms: 0.3333333333 + -0.3333333333 = 0.0000000000
0.0000000000 + -4x2 + x4 = 0 + -0.3333333333
-4x2 + x4 = 0 + -0.3333333333

Combine like terms: 0 + -0.3333333333 = -0.3333333333
-4x2 + x4 = -0.3333333333

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 = -0.3333333333 + 4

Reorder the terms:
4 + -4x2 + x4 = -0.3333333333 + 4

Combine like terms: -0.3333333333 + 4 = 3.6666666667
4 + -4x2 + x4 = 3.6666666667

Factor a perfect square on the left side:
(x2 + -2)(x2 + -2) = 3.6666666667

Calculate the square root of the right side: 1.914854216

Break this problem into two subproblems by setting 
(x2 + -2) equal to 1.914854216 and -1.914854216.

Subproblem 1

x2 + -2 = 1.914854216 Simplifying x2 + -2 = 1.914854216 Reorder the terms: -2 + x2 = 1.914854216 Solving -2 + x2 = 1.914854216 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 = 1.914854216 + 2 Combine like terms: -2 + 2 = 0 0 + x2 = 1.914854216 + 2 x2 = 1.914854216 + 2 Combine like terms: 1.914854216 + 2 = 3.914854216 x2 = 3.914854216 Simplifying x2 = 3.914854216 Take the square root of each side: x = {-1.978599054, 1.978599054}

Subproblem 2

x2 + -2 = -1.914854216 Simplifying x2 + -2 = -1.914854216 Reorder the terms: -2 + x2 = -1.914854216 Solving -2 + x2 = -1.914854216 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 = -1.914854216 + 2 Combine like terms: -2 + 2 = 0 0 + x2 = -1.914854216 + 2 x2 = -1.914854216 + 2 Combine like terms: -1.914854216 + 2 = 0.085145784 x2 = 0.085145784 Simplifying x2 = 0.085145784 Take the square root of each side: x = {-0.291797505, 0.291797505}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-1.978599054, 1.978599054, -0.291797505, 0.291797505}

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