16x^4-81x^2+1=0

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Solution for 16x^4-81x^2+1=0 equation:


Simplifying
16x4 + -81x2 + 1 = 0

Reorder the terms:
1 + -81x2 + 16x4 = 0

Solving
1 + -81x2 + 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 + -5.0625x2 + x4 = 0

Move the constant term to the right:

Add '-0.0625' to each side of the equation.
0.0625 + -5.0625x2 + -0.0625 + x4 = 0 + -0.0625

Reorder the terms:
0.0625 + -0.0625 + -5.0625x2 + x4 = 0 + -0.0625

Combine like terms: 0.0625 + -0.0625 = 0.0000
0.0000 + -5.0625x2 + x4 = 0 + -0.0625
-5.0625x2 + x4 = 0 + -0.0625

Combine like terms: 0 + -0.0625 = -0.0625
-5.0625x2 + x4 = -0.0625

The x term is -5.0625x2.  Take half its coefficient (-2.53125).
Square it (6.407226563) and add it to both sides.

Add '6.407226563' to each side of the equation.
-5.0625x2 + 6.407226563 + x4 = -0.0625 + 6.407226563

Reorder the terms:
6.407226563 + -5.0625x2 + x4 = -0.0625 + 6.407226563

Combine like terms: -0.0625 + 6.407226563 = 6.344726563
6.407226563 + -5.0625x2 + x4 = 6.344726563

Factor a perfect square on the left side:
(x2 + -2.53125)(x2 + -2.53125) = 6.344726563

Calculate the square root of the right side: 2.518874067

Break this problem into two subproblems by setting 
(x2 + -2.53125) equal to 2.518874067 and -2.518874067.

Subproblem 1

x2 + -2.53125 = 2.518874067 Simplifying x2 + -2.53125 = 2.518874067 Reorder the terms: -2.53125 + x2 = 2.518874067 Solving -2.53125 + x2 = 2.518874067 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.53125' to each side of the equation. -2.53125 + 2.53125 + x2 = 2.518874067 + 2.53125 Combine like terms: -2.53125 + 2.53125 = 0.00000 0.00000 + x2 = 2.518874067 + 2.53125 x2 = 2.518874067 + 2.53125 Combine like terms: 2.518874067 + 2.53125 = 5.050124067 x2 = 5.050124067 Simplifying x2 = 5.050124067 Take the square root of each side: x = {-2.24724811, 2.24724811}

Subproblem 2

x2 + -2.53125 = -2.518874067 Simplifying x2 + -2.53125 = -2.518874067 Reorder the terms: -2.53125 + x2 = -2.518874067 Solving -2.53125 + x2 = -2.518874067 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.53125' to each side of the equation. -2.53125 + 2.53125 + x2 = -2.518874067 + 2.53125 Combine like terms: -2.53125 + 2.53125 = 0.00000 0.00000 + x2 = -2.518874067 + 2.53125 x2 = -2.518874067 + 2.53125 Combine like terms: -2.518874067 + 2.53125 = 0.012375933 x2 = 0.012375933 Simplifying x2 = 0.012375933 Take the square root of each side: x = {-0.111247171, 0.111247171}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-2.24724811, 2.24724811, -0.111247171, 0.111247171}

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