72x^4-32x^2+1=0

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Solution for 72x^4-32x^2+1=0 equation:


Simplifying
72x4 + -32x2 + 1 = 0

Reorder the terms:
1 + -32x2 + 72x4 = 0

Solving
1 + -32x2 + 72x4 = 0

Solving for variable 'x'.

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

Divide each side by '72'.
0.01388888889 + -0.4444444444x2 + x4 = 0

Move the constant term to the right:

Add '-0.01388888889' to each side of the equation.
0.01388888889 + -0.4444444444x2 + -0.01388888889 + x4 = 0 + -0.01388888889

Reorder the terms:
0.01388888889 + -0.01388888889 + -0.4444444444x2 + x4 = 0 + -0.01388888889

Combine like terms: 0.01388888889 + -0.01388888889 = 0.00000000000
0.00000000000 + -0.4444444444x2 + x4 = 0 + -0.01388888889
-0.4444444444x2 + x4 = 0 + -0.01388888889

Combine like terms: 0 + -0.01388888889 = -0.01388888889
-0.4444444444x2 + x4 = -0.01388888889

The x term is -0.4444444444x2.  Take half its coefficient (-0.2222222222).
Square it (0.04938271604) and add it to both sides.

Add '0.04938271604' to each side of the equation.
-0.4444444444x2 + 0.04938271604 + x4 = -0.01388888889 + 0.04938271604

Reorder the terms:
0.04938271604 + -0.4444444444x2 + x4 = -0.01388888889 + 0.04938271604

Combine like terms: -0.01388888889 + 0.04938271604 = 0.03549382715
0.04938271604 + -0.4444444444x2 + x4 = 0.03549382715

Factor a perfect square on the left side:
(x2 + -0.2222222222)(x2 + -0.2222222222) = 0.03549382715

Calculate the square root of the right side: 0.188398055

Break this problem into two subproblems by setting 
(x2 + -0.2222222222) equal to 0.188398055 and -0.188398055.

Subproblem 1

x2 + -0.2222222222 = 0.188398055 Simplifying x2 + -0.2222222222 = 0.188398055 Reorder the terms: -0.2222222222 + x2 = 0.188398055 Solving -0.2222222222 + x2 = 0.188398055 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.2222222222' to each side of the equation. -0.2222222222 + 0.2222222222 + x2 = 0.188398055 + 0.2222222222 Combine like terms: -0.2222222222 + 0.2222222222 = 0.0000000000 0.0000000000 + x2 = 0.188398055 + 0.2222222222 x2 = 0.188398055 + 0.2222222222 Combine like terms: 0.188398055 + 0.2222222222 = 0.4106202772 x2 = 0.4106202772 Simplifying x2 = 0.4106202772 Take the square root of each side: x = {-0.640796596, 0.640796596}

Subproblem 2

x2 + -0.2222222222 = -0.188398055 Simplifying x2 + -0.2222222222 = -0.188398055 Reorder the terms: -0.2222222222 + x2 = -0.188398055 Solving -0.2222222222 + x2 = -0.188398055 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.2222222222' to each side of the equation. -0.2222222222 + 0.2222222222 + x2 = -0.188398055 + 0.2222222222 Combine like terms: -0.2222222222 + 0.2222222222 = 0.0000000000 0.0000000000 + x2 = -0.188398055 + 0.2222222222 x2 = -0.188398055 + 0.2222222222 Combine like terms: -0.188398055 + 0.2222222222 = 0.0338241672 x2 = 0.0338241672 Simplifying x2 = 0.0338241672 Take the square root of each side: x = {-0.183913477, 0.183913477}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.640796596, 0.640796596, -0.183913477, 0.183913477}

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