64x^4-320x^2+1=0

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Solution for 64x^4-320x^2+1=0 equation:


Simplifying
64x4 + -320x2 + 1 = 0

Reorder the terms:
1 + -320x2 + 64x4 = 0

Solving
1 + -320x2 + 64x4 = 0

Solving for variable 'x'.

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

Divide each side by '64'.
0.015625 + -5x2 + x4 = 0

Move the constant term to the right:

Add '-0.015625' to each side of the equation.
0.015625 + -5x2 + -0.015625 + x4 = 0 + -0.015625

Reorder the terms:
0.015625 + -0.015625 + -5x2 + x4 = 0 + -0.015625

Combine like terms: 0.015625 + -0.015625 = 0.000000
0.000000 + -5x2 + x4 = 0 + -0.015625
-5x2 + x4 = 0 + -0.015625

Combine like terms: 0 + -0.015625 = -0.015625
-5x2 + x4 = -0.015625

The x term is -5x2.  Take half its coefficient (-2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
-5x2 + 6.25 + x4 = -0.015625 + 6.25

Reorder the terms:
6.25 + -5x2 + x4 = -0.015625 + 6.25

Combine like terms: -0.015625 + 6.25 = 6.234375
6.25 + -5x2 + x4 = 6.234375

Factor a perfect square on the left side:
(x2 + -2.5)(x2 + -2.5) = 6.234375

Calculate the square root of the right side: 2.496873044

Break this problem into two subproblems by setting 
(x2 + -2.5) equal to 2.496873044 and -2.496873044.

Subproblem 1

x2 + -2.5 = 2.496873044 Simplifying x2 + -2.5 = 2.496873044 Reorder the terms: -2.5 + x2 = 2.496873044 Solving -2.5 + x2 = 2.496873044 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + x2 = 2.496873044 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + x2 = 2.496873044 + 2.5 x2 = 2.496873044 + 2.5 Combine like terms: 2.496873044 + 2.5 = 4.996873044 x2 = 4.996873044 Simplifying x2 = 4.996873044 Take the square root of each side: x = {-2.23536866, 2.23536866}

Subproblem 2

x2 + -2.5 = -2.496873044 Simplifying x2 + -2.5 = -2.496873044 Reorder the terms: -2.5 + x2 = -2.496873044 Solving -2.5 + x2 = -2.496873044 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + x2 = -2.496873044 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + x2 = -2.496873044 + 2.5 x2 = -2.496873044 + 2.5 Combine like terms: -2.496873044 + 2.5 = 0.003126956 x2 = 0.003126956 Simplifying x2 = 0.003126956 Take the square root of each side: x = {-0.055919192, 0.055919192}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-2.23536866, 2.23536866, -0.055919192, 0.055919192}

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