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

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


Simplifying
x4 + -16x2 + 1 = 0

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

Solving
1 + -16x2 + x4 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + -16x2 + -1 + x4 = 0 + -1

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

Combine like terms: 1 + -1 = 0
0 + -16x2 + x4 = 0 + -1
-16x2 + x4 = 0 + -1

Combine like terms: 0 + -1 = -1
-16x2 + x4 = -1

The x term is -16x2.  Take half its coefficient (-8).
Square it (64) and add it to both sides.

Add '64' to each side of the equation.
-16x2 + 64 + x4 = -1 + 64

Reorder the terms:
64 + -16x2 + x4 = -1 + 64

Combine like terms: -1 + 64 = 63
64 + -16x2 + x4 = 63

Factor a perfect square on the left side:
(x2 + -8)(x2 + -8) = 63

Calculate the square root of the right side: 7.937253933

Break this problem into two subproblems by setting 
(x2 + -8) equal to 7.937253933 and -7.937253933.

Subproblem 1

x2 + -8 = 7.937253933 Simplifying x2 + -8 = 7.937253933 Reorder the terms: -8 + x2 = 7.937253933 Solving -8 + x2 = 7.937253933 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '8' to each side of the equation. -8 + 8 + x2 = 7.937253933 + 8 Combine like terms: -8 + 8 = 0 0 + x2 = 7.937253933 + 8 x2 = 7.937253933 + 8 Combine like terms: 7.937253933 + 8 = 15.937253933 x2 = 15.937253933 Simplifying x2 = 15.937253933 Take the square root of each side: x = {-3.992149037, 3.992149037}

Subproblem 2

x2 + -8 = -7.937253933 Simplifying x2 + -8 = -7.937253933 Reorder the terms: -8 + x2 = -7.937253933 Solving -8 + x2 = -7.937253933 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '8' to each side of the equation. -8 + 8 + x2 = -7.937253933 + 8 Combine like terms: -8 + 8 = 0 0 + x2 = -7.937253933 + 8 x2 = -7.937253933 + 8 Combine like terms: -7.937253933 + 8 = 0.062746067 x2 = 0.062746067 Simplifying x2 = 0.062746067 Take the square root of each side: x = {-0.250491651, 0.250491651}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-3.992149037, 3.992149037, -0.250491651, 0.250491651}

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