0=x^4-18x^2+6

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Solution for 0=x^4-18x^2+6 equation:


Simplifying
0 = x4 + -18x2 + 6

Reorder the terms:
0 = 6 + -18x2 + x4

Solving
0 = 6 + -18x2 + x4

Solving for variable 'x'.

Combine like terms: 0 + -6 = -6
-6 + 18x2 + -1x4 = 6 + -18x2 + x4 + -6 + 18x2 + -1x4

Reorder the terms:
-6 + 18x2 + -1x4 = 6 + -6 + -18x2 + 18x2 + x4 + -1x4

Combine like terms: 6 + -6 = 0
-6 + 18x2 + -1x4 = 0 + -18x2 + 18x2 + x4 + -1x4
-6 + 18x2 + -1x4 = -18x2 + 18x2 + x4 + -1x4

Combine like terms: -18x2 + 18x2 = 0
-6 + 18x2 + -1x4 = 0 + x4 + -1x4
-6 + 18x2 + -1x4 = x4 + -1x4

Combine like terms: x4 + -1x4 = 0
-6 + 18x2 + -1x4 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
6 + -18x2 + x4 = 0

Move the constant term to the right:

Add '-6' to each side of the equation.
6 + -18x2 + -6 + x4 = 0 + -6

Reorder the terms:
6 + -6 + -18x2 + x4 = 0 + -6

Combine like terms: 6 + -6 = 0
0 + -18x2 + x4 = 0 + -6
-18x2 + x4 = 0 + -6

Combine like terms: 0 + -6 = -6
-18x2 + x4 = -6

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 = -6 + 81

Reorder the terms:
81 + -18x2 + x4 = -6 + 81

Combine like terms: -6 + 81 = 75
81 + -18x2 + x4 = 75

Factor a perfect square on the left side:
(x2 + -9)(x2 + -9) = 75

Calculate the square root of the right side: 8.660254038

Break this problem into two subproblems by setting 
(x2 + -9) equal to 8.660254038 and -8.660254038.

Subproblem 1

x2 + -9 = 8.660254038 Simplifying x2 + -9 = 8.660254038 Reorder the terms: -9 + x2 = 8.660254038 Solving -9 + x2 = 8.660254038 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 = 8.660254038 + 9 Combine like terms: -9 + 9 = 0 0 + x2 = 8.660254038 + 9 x2 = 8.660254038 + 9 Combine like terms: 8.660254038 + 9 = 17.660254038 x2 = 17.660254038 Simplifying x2 = 17.660254038 Take the square root of each side: x = {-4.202410503, 4.202410503}

Subproblem 2

x2 + -9 = -8.660254038 Simplifying x2 + -9 = -8.660254038 Reorder the terms: -9 + x2 = -8.660254038 Solving -9 + x2 = -8.660254038 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 = -8.660254038 + 9 Combine like terms: -9 + 9 = 0 0 + x2 = -8.660254038 + 9 x2 = -8.660254038 + 9 Combine like terms: -8.660254038 + 9 = 0.339745962 x2 = 0.339745962 Simplifying x2 = 0.339745962 Take the square root of each side: x = {-0.582877313, 0.582877313}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-4.202410503, 4.202410503, -0.582877313, 0.582877313}

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