0=-36x^4+36x^2-1

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Solution for 0=-36x^4+36x^2-1 equation:


Simplifying
0 = -36x4 + 36x2 + -1

Reorder the terms:
0 = -1 + 36x2 + -36x4

Solving
0 = -1 + 36x2 + -36x4

Solving for variable 'x'.

Combine like terms: 0 + 1 = 1
1 + -36x2 + 36x4 = -1 + 36x2 + -36x4 + 1 + -36x2 + 36x4

Reorder the terms:
1 + -36x2 + 36x4 = -1 + 1 + 36x2 + -36x2 + -36x4 + 36x4

Combine like terms: -1 + 1 = 0
1 + -36x2 + 36x4 = 0 + 36x2 + -36x2 + -36x4 + 36x4
1 + -36x2 + 36x4 = 36x2 + -36x2 + -36x4 + 36x4

Combine like terms: 36x2 + -36x2 = 0
1 + -36x2 + 36x4 = 0 + -36x4 + 36x4
1 + -36x2 + 36x4 = -36x4 + 36x4

Combine like terms: -36x4 + 36x4 = 0
1 + -36x2 + 36x4 = 0

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

Divide each side by '36'.
0.02777777778 + -1x2 + x4 = 0

Move the constant term to the right:

Add '-0.02777777778' to each side of the equation.
0.02777777778 + -1x2 + -0.02777777778 + x4 = 0 + -0.02777777778

Reorder the terms:
0.02777777778 + -0.02777777778 + -1x2 + x4 = 0 + -0.02777777778

Combine like terms: 0.02777777778 + -0.02777777778 = 0.00000000000
0.00000000000 + -1x2 + x4 = 0 + -0.02777777778
-1x2 + x4 = 0 + -0.02777777778

Combine like terms: 0 + -0.02777777778 = -0.02777777778
-1x2 + x4 = -0.02777777778

The x term is -1x2.  Take half its coefficient (-0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1x2 + 0.25 + x4 = -0.02777777778 + 0.25

Reorder the terms:
0.25 + -1x2 + x4 = -0.02777777778 + 0.25

Combine like terms: -0.02777777778 + 0.25 = 0.22222222222
0.25 + -1x2 + x4 = 0.22222222222

Factor a perfect square on the left side:
(x2 + -0.5)(x2 + -0.5) = 0.22222222222

Calculate the square root of the right side: 0.471404521

Break this problem into two subproblems by setting 
(x2 + -0.5) equal to 0.471404521 and -0.471404521.

Subproblem 1

x2 + -0.5 = 0.471404521 Simplifying x2 + -0.5 = 0.471404521 Reorder the terms: -0.5 + x2 = 0.471404521 Solving -0.5 + x2 = 0.471404521 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x2 = 0.471404521 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x2 = 0.471404521 + 0.5 x2 = 0.471404521 + 0.5 Combine like terms: 0.471404521 + 0.5 = 0.971404521 x2 = 0.971404521 Simplifying x2 = 0.971404521 Take the square root of each side: x = {-0.98559856, 0.98559856}

Subproblem 2

x2 + -0.5 = -0.471404521 Simplifying x2 + -0.5 = -0.471404521 Reorder the terms: -0.5 + x2 = -0.471404521 Solving -0.5 + x2 = -0.471404521 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x2 = -0.471404521 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x2 = -0.471404521 + 0.5 x2 = -0.471404521 + 0.5 Combine like terms: -0.471404521 + 0.5 = 0.028595479 x2 = 0.028595479 Simplifying x2 = 0.028595479 Take the square root of each side: x = {-0.169101978, 0.169101978}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.98559856, 0.98559856, -0.169101978, 0.169101978}

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