0=x^4-18x^2+9

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Solution for 0=x^4-18x^2+9 equation:


Simplifying
0 = x4 + -18x2 + 9

Reorder the terms:
0 = 9 + -18x2 + x4

Solving
0 = 9 + -18x2 + x4

Solving for variable 'x'.

Combine like terms: 0 + -9 = -9
-9 + 18x2 + -1x4 = 9 + -18x2 + x4 + -9 + 18x2 + -1x4

Reorder the terms:
-9 + 18x2 + -1x4 = 9 + -9 + -18x2 + 18x2 + x4 + -1x4

Combine like terms: 9 + -9 = 0
-9 + 18x2 + -1x4 = 0 + -18x2 + 18x2 + x4 + -1x4
-9 + 18x2 + -1x4 = -18x2 + 18x2 + x4 + -1x4

Combine like terms: -18x2 + 18x2 = 0
-9 + 18x2 + -1x4 = 0 + x4 + -1x4
-9 + 18x2 + -1x4 = x4 + -1x4

Combine like terms: x4 + -1x4 = 0
-9 + 18x2 + -1x4 = 0

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

Divide each side by '-1'.
9 + -18x2 + x4 = 0

Move the constant term to the right:

Add '-9' to each side of the equation.
9 + -18x2 + -9 + x4 = 0 + -9

Reorder the terms:
9 + -9 + -18x2 + x4 = 0 + -9

Combine like terms: 9 + -9 = 0
0 + -18x2 + x4 = 0 + -9
-18x2 + x4 = 0 + -9

Combine like terms: 0 + -9 = -9
-18x2 + x4 = -9

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 = -9 + 81

Reorder the terms:
81 + -18x2 + x4 = -9 + 81

Combine like terms: -9 + 81 = 72
81 + -18x2 + x4 = 72

Factor a perfect square on the left side:
(x2 + -9)(x2 + -9) = 72

Calculate the square root of the right side: 8.485281374

Break this problem into two subproblems by setting 
(x2 + -9) equal to 8.485281374 and -8.485281374.

Subproblem 1

x2 + -9 = 8.485281374 Simplifying x2 + -9 = 8.485281374 Reorder the terms: -9 + x2 = 8.485281374 Solving -9 + x2 = 8.485281374 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 = 8.485281374 + 9 Combine like terms: -9 + 9 = 0 0 + x2 = 8.485281374 + 9 x2 = 8.485281374 + 9 Combine like terms: 8.485281374 + 9 = 17.485281374 x2 = 17.485281374 Simplifying x2 = 17.485281374 Take the square root of each side: x = {-4.18154055, 4.18154055}

Subproblem 2

x2 + -9 = -8.485281374 Simplifying x2 + -9 = -8.485281374 Reorder the terms: -9 + x2 = -8.485281374 Solving -9 + x2 = -8.485281374 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 = -8.485281374 + 9 Combine like terms: -9 + 9 = 0 0 + x2 = -8.485281374 + 9 x2 = -8.485281374 + 9 Combine like terms: -8.485281374 + 9 = 0.514718626 x2 = 0.514718626 Simplifying x2 = 0.514718626 Take the square root of each side: x = {-0.717438935, 0.717438935}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-4.18154055, 4.18154055, -0.717438935, 0.717438935}

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