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

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


Simplifying
0 = 4x4 + -12x2 + 3

Reorder the terms:
0 = 3 + -12x2 + 4x4

Solving
0 = 3 + -12x2 + 4x4

Solving for variable 'x'.

Combine like terms: 0 + -3 = -3
-3 + 12x2 + -4x4 = 3 + -12x2 + 4x4 + -3 + 12x2 + -4x4

Reorder the terms:
-3 + 12x2 + -4x4 = 3 + -3 + -12x2 + 12x2 + 4x4 + -4x4

Combine like terms: 3 + -3 = 0
-3 + 12x2 + -4x4 = 0 + -12x2 + 12x2 + 4x4 + -4x4
-3 + 12x2 + -4x4 = -12x2 + 12x2 + 4x4 + -4x4

Combine like terms: -12x2 + 12x2 = 0
-3 + 12x2 + -4x4 = 0 + 4x4 + -4x4
-3 + 12x2 + -4x4 = 4x4 + -4x4

Combine like terms: 4x4 + -4x4 = 0
-3 + 12x2 + -4x4 = 0

Begin completing the square.  Divide all terms by
-4 the coefficient of the squared term: 

Divide each side by '-4'.
0.75 + -3x2 + x4 = 0

Move the constant term to the right:

Add '-0.75' to each side of the equation.
0.75 + -3x2 + -0.75 + x4 = 0 + -0.75

Reorder the terms:
0.75 + -0.75 + -3x2 + x4 = 0 + -0.75

Combine like terms: 0.75 + -0.75 = 0.00
0.00 + -3x2 + x4 = 0 + -0.75
-3x2 + x4 = 0 + -0.75

Combine like terms: 0 + -0.75 = -0.75
-3x2 + x4 = -0.75

The x term is -3x2.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3x2 + 2.25 + x4 = -0.75 + 2.25

Reorder the terms:
2.25 + -3x2 + x4 = -0.75 + 2.25

Combine like terms: -0.75 + 2.25 = 1.5
2.25 + -3x2 + x4 = 1.5

Factor a perfect square on the left side:
(x2 + -1.5)(x2 + -1.5) = 1.5

Calculate the square root of the right side: 1.224744871

Break this problem into two subproblems by setting 
(x2 + -1.5) equal to 1.224744871 and -1.224744871.

Subproblem 1

x2 + -1.5 = 1.224744871 Simplifying x2 + -1.5 = 1.224744871 Reorder the terms: -1.5 + x2 = 1.224744871 Solving -1.5 + x2 = 1.224744871 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + x2 = 1.224744871 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x2 = 1.224744871 + 1.5 x2 = 1.224744871 + 1.5 Combine like terms: 1.224744871 + 1.5 = 2.724744871 x2 = 2.724744871 Simplifying x2 = 2.724744871 Take the square root of each side: x = {-1.650680124, 1.650680124}

Subproblem 2

x2 + -1.5 = -1.224744871 Simplifying x2 + -1.5 = -1.224744871 Reorder the terms: -1.5 + x2 = -1.224744871 Solving -1.5 + x2 = -1.224744871 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + x2 = -1.224744871 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x2 = -1.224744871 + 1.5 x2 = -1.224744871 + 1.5 Combine like terms: -1.224744871 + 1.5 = 0.275255129 x2 = 0.275255129 Simplifying x2 = 0.275255129 Take the square root of each side: x = {-0.524647624, 0.524647624}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-1.650680124, 1.650680124, -0.524647624, 0.524647624}

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