3x^4-37x^2+36=0

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Solution for 3x^4-37x^2+36=0 equation:


Simplifying
3x4 + -37x2 + 36 = 0

Reorder the terms:
36 + -37x2 + 3x4 = 0

Solving
36 + -37x2 + 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'.
12 + -12.33333333x2 + x4 = 0

Move the constant term to the right:

Add '-12' to each side of the equation.
12 + -12.33333333x2 + -12 + x4 = 0 + -12

Reorder the terms:
12 + -12 + -12.33333333x2 + x4 = 0 + -12

Combine like terms: 12 + -12 = 0
0 + -12.33333333x2 + x4 = 0 + -12
-12.33333333x2 + x4 = 0 + -12

Combine like terms: 0 + -12 = -12
-12.33333333x2 + x4 = -12

The x term is -12.33333333x2.  Take half its coefficient (-6.166666665).
Square it (38.02777776) and add it to both sides.

Add '38.02777776' to each side of the equation.
-12.33333333x2 + 38.02777776 + x4 = -12 + 38.02777776

Reorder the terms:
38.02777776 + -12.33333333x2 + x4 = -12 + 38.02777776

Combine like terms: -12 + 38.02777776 = 26.02777776
38.02777776 + -12.33333333x2 + x4 = 26.02777776

Factor a perfect square on the left side:
(x2 + -6.166666665)(x2 + -6.166666665) = 26.02777776

Calculate the square root of the right side: 5.10174262

Break this problem into two subproblems by setting 
(x2 + -6.166666665) equal to 5.10174262 and -5.10174262.

Subproblem 1

x2 + -6.166666665 = 5.10174262 Simplifying x2 + -6.166666665 = 5.10174262 Reorder the terms: -6.166666665 + x2 = 5.10174262 Solving -6.166666665 + x2 = 5.10174262 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '6.166666665' to each side of the equation. -6.166666665 + 6.166666665 + x2 = 5.10174262 + 6.166666665 Combine like terms: -6.166666665 + 6.166666665 = 0.000000000 0.000000000 + x2 = 5.10174262 + 6.166666665 x2 = 5.10174262 + 6.166666665 Combine like terms: 5.10174262 + 6.166666665 = 11.268409285 x2 = 11.268409285 Simplifying x2 = 11.268409285 Take the square root of each side: x = {-3.356845139, 3.356845139}

Subproblem 2

x2 + -6.166666665 = -5.10174262 Simplifying x2 + -6.166666665 = -5.10174262 Reorder the terms: -6.166666665 + x2 = -5.10174262 Solving -6.166666665 + x2 = -5.10174262 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '6.166666665' to each side of the equation. -6.166666665 + 6.166666665 + x2 = -5.10174262 + 6.166666665 Combine like terms: -6.166666665 + 6.166666665 = 0.000000000 0.000000000 + x2 = -5.10174262 + 6.166666665 x2 = -5.10174262 + 6.166666665 Combine like terms: -5.10174262 + 6.166666665 = 1.064924045 x2 = 1.064924045 Simplifying x2 = 1.064924045 Take the square root of each side: x = {-1.031951571, 1.031951571}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-3.356845139, 3.356845139, -1.031951571, 1.031951571}

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