16x^4-20x^2+1=0

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Solution for 16x^4-20x^2+1=0 equation:


Simplifying
16x4 + -20x2 + 1 = 0

Reorder the terms:
1 + -20x2 + 16x4 = 0

Solving
1 + -20x2 + 16x4 = 0

Solving for variable 'x'.

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

Divide each side by '16'.
0.0625 + -1.25x2 + x4 = 0

Move the constant term to the right:

Add '-0.0625' to each side of the equation.
0.0625 + -1.25x2 + -0.0625 + x4 = 0 + -0.0625

Reorder the terms:
0.0625 + -0.0625 + -1.25x2 + x4 = 0 + -0.0625

Combine like terms: 0.0625 + -0.0625 = 0.0000
0.0000 + -1.25x2 + x4 = 0 + -0.0625
-1.25x2 + x4 = 0 + -0.0625

Combine like terms: 0 + -0.0625 = -0.0625
-1.25x2 + x4 = -0.0625

The x term is -1.25x2.  Take half its coefficient (-0.625).
Square it (0.390625) and add it to both sides.

Add '0.390625' to each side of the equation.
-1.25x2 + 0.390625 + x4 = -0.0625 + 0.390625

Reorder the terms:
0.390625 + -1.25x2 + x4 = -0.0625 + 0.390625

Combine like terms: -0.0625 + 0.390625 = 0.328125
0.390625 + -1.25x2 + x4 = 0.328125

Factor a perfect square on the left side:
(x2 + -0.625)(x2 + -0.625) = 0.328125

Calculate the square root of the right side: 0.572821962

Break this problem into two subproblems by setting 
(x2 + -0.625) equal to 0.572821962 and -0.572821962.

Subproblem 1

x2 + -0.625 = 0.572821962 Simplifying x2 + -0.625 = 0.572821962 Reorder the terms: -0.625 + x2 = 0.572821962 Solving -0.625 + x2 = 0.572821962 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.625' to each side of the equation. -0.625 + 0.625 + x2 = 0.572821962 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + x2 = 0.572821962 + 0.625 x2 = 0.572821962 + 0.625 Combine like terms: 0.572821962 + 0.625 = 1.197821962 x2 = 1.197821962 Simplifying x2 = 1.197821962 Take the square root of each side: x = {-1.09445053, 1.09445053}

Subproblem 2

x2 + -0.625 = -0.572821962 Simplifying x2 + -0.625 = -0.572821962 Reorder the terms: -0.625 + x2 = -0.572821962 Solving -0.625 + x2 = -0.572821962 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.625' to each side of the equation. -0.625 + 0.625 + x2 = -0.572821962 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + x2 = -0.572821962 + 0.625 x2 = -0.572821962 + 0.625 Combine like terms: -0.572821962 + 0.625 = 0.052178038 x2 = 0.052178038 Simplifying x2 = 0.052178038 Take the square root of each side: x = {-0.228425126, 0.228425126}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-1.09445053, 1.09445053, -0.228425126, 0.228425126}

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