3x^4-16x^2+3=0

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Solution for 3x^4-16x^2+3=0 equation:


Simplifying
3x4 + -16x2 + 3 = 0

Reorder the terms:
3 + -16x2 + 3x4 = 0

Solving
3 + -16x2 + 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'.
1 + -5.333333333x2 + x4 = 0

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + -5.333333333x2 + -1 + x4 = 0 + -1

Reorder the terms:
1 + -1 + -5.333333333x2 + x4 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + -5.333333333x2 + x4 = 0 + -1
-5.333333333x2 + x4 = 0 + -1

Combine like terms: 0 + -1 = -1
-5.333333333x2 + x4 = -1

The x term is -5.333333333x2.  Take half its coefficient (-2.666666667).
Square it (7.111111113) and add it to both sides.

Add '7.111111113' to each side of the equation.
-5.333333333x2 + 7.111111113 + x4 = -1 + 7.111111113

Reorder the terms:
7.111111113 + -5.333333333x2 + x4 = -1 + 7.111111113

Combine like terms: -1 + 7.111111113 = 6.111111113
7.111111113 + -5.333333333x2 + x4 = 6.111111113

Factor a perfect square on the left side:
(x2 + -2.666666667)(x2 + -2.666666667) = 6.111111113

Calculate the square root of the right side: 2.472066163

Break this problem into two subproblems by setting 
(x2 + -2.666666667) equal to 2.472066163 and -2.472066163.

Subproblem 1

x2 + -2.666666667 = 2.472066163 Simplifying x2 + -2.666666667 = 2.472066163 Reorder the terms: -2.666666667 + x2 = 2.472066163 Solving -2.666666667 + x2 = 2.472066163 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.666666667' to each side of the equation. -2.666666667 + 2.666666667 + x2 = 2.472066163 + 2.666666667 Combine like terms: -2.666666667 + 2.666666667 = 0.000000000 0.000000000 + x2 = 2.472066163 + 2.666666667 x2 = 2.472066163 + 2.666666667 Combine like terms: 2.472066163 + 2.666666667 = 5.13873283 x2 = 5.13873283 Simplifying x2 = 5.13873283 Take the square root of each side: x = {-2.26687733, 2.26687733}

Subproblem 2

x2 + -2.666666667 = -2.472066163 Simplifying x2 + -2.666666667 = -2.472066163 Reorder the terms: -2.666666667 + x2 = -2.472066163 Solving -2.666666667 + x2 = -2.472066163 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.666666667' to each side of the equation. -2.666666667 + 2.666666667 + x2 = -2.472066163 + 2.666666667 Combine like terms: -2.666666667 + 2.666666667 = 0.000000000 0.000000000 + x2 = -2.472066163 + 2.666666667 x2 = -2.472066163 + 2.666666667 Combine like terms: -2.472066163 + 2.666666667 = 0.194600504 x2 = 0.194600504 Simplifying x2 = 0.194600504 Take the square root of each side: x = {-0.441135471, 0.441135471}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-2.26687733, 2.26687733, -0.441135471, 0.441135471}

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