x^4-18x^2+1=0

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Solution for x^4-18x^2+1=0 equation:


Simplifying
x4 + -18x2 + 1 = 0

Reorder the terms:
1 + -18x2 + x4 = 0

Solving
1 + -18x2 + 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 + -18x2 + -1 + x4 = 0 + -1

Reorder the terms:
1 + -1 + -18x2 + x4 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + -18x2 + x4 = 0 + -1
-18x2 + x4 = 0 + -1

Combine like terms: 0 + -1 = -1
-18x2 + x4 = -1

The x term is -18x2.  Take half its coefficient (-9).
Square it (81) and add it to both sides.

Add '81' to each side of the equation.
-18x2 + 81 + x4 = -1 + 81

Reorder the terms:
81 + -18x2 + x4 = -1 + 81

Combine like terms: -1 + 81 = 80
81 + -18x2 + x4 = 80

Factor a perfect square on the left side:
(x2 + -9)(x2 + -9) = 80

Calculate the square root of the right side: 8.94427191

Break this problem into two subproblems by setting 
(x2 + -9) equal to 8.94427191 and -8.94427191.

Subproblem 1

x2 + -9 = 8.94427191 Simplifying x2 + -9 = 8.94427191 Reorder the terms: -9 + x2 = 8.94427191 Solving -9 + x2 = 8.94427191 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '9' to each side of the equation. -9 + 9 + x2 = 8.94427191 + 9 Combine like terms: -9 + 9 = 0 0 + x2 = 8.94427191 + 9 x2 = 8.94427191 + 9 Combine like terms: 8.94427191 + 9 = 17.94427191 x2 = 17.94427191 Simplifying x2 = 17.94427191 Take the square root of each side: x = {-4.236067978, 4.236067978}

Subproblem 2

x2 + -9 = -8.94427191 Simplifying x2 + -9 = -8.94427191 Reorder the terms: -9 + x2 = -8.94427191 Solving -9 + x2 = -8.94427191 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '9' to each side of the equation. -9 + 9 + x2 = -8.94427191 + 9 Combine like terms: -9 + 9 = 0 0 + x2 = -8.94427191 + 9 x2 = -8.94427191 + 9 Combine like terms: -8.94427191 + 9 = 0.05572809 x2 = 0.05572809 Simplifying x2 = 0.05572809 Take the square root of each side: x = {-0.236067977, 0.236067977}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-4.236067978, 4.236067978, -0.236067977, 0.236067977}

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