x^4-12x^2+18=0

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Solution for x^4-12x^2+18=0 equation:


Simplifying
x4 + -12x2 + 18 = 0

Reorder the terms:
18 + -12x2 + x4 = 0

Solving
18 + -12x2 + x4 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-18' to each side of the equation.
18 + -12x2 + -18 + x4 = 0 + -18

Reorder the terms:
18 + -18 + -12x2 + x4 = 0 + -18

Combine like terms: 18 + -18 = 0
0 + -12x2 + x4 = 0 + -18
-12x2 + x4 = 0 + -18

Combine like terms: 0 + -18 = -18
-12x2 + x4 = -18

The x term is -12x2.  Take half its coefficient (-6).
Square it (36) and add it to both sides.

Add '36' to each side of the equation.
-12x2 + 36 + x4 = -18 + 36

Reorder the terms:
36 + -12x2 + x4 = -18 + 36

Combine like terms: -18 + 36 = 18
36 + -12x2 + x4 = 18

Factor a perfect square on the left side:
(x2 + -6)(x2 + -6) = 18

Calculate the square root of the right side: 4.242640687

Break this problem into two subproblems by setting 
(x2 + -6) equal to 4.242640687 and -4.242640687.

Subproblem 1

x2 + -6 = 4.242640687 Simplifying x2 + -6 = 4.242640687 Reorder the terms: -6 + x2 = 4.242640687 Solving -6 + x2 = 4.242640687 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '6' to each side of the equation. -6 + 6 + x2 = 4.242640687 + 6 Combine like terms: -6 + 6 = 0 0 + x2 = 4.242640687 + 6 x2 = 4.242640687 + 6 Combine like terms: 4.242640687 + 6 = 10.242640687 x2 = 10.242640687 Simplifying x2 = 10.242640687 Take the square root of each side: x = {-3.200412581, 3.200412581}

Subproblem 2

x2 + -6 = -4.242640687 Simplifying x2 + -6 = -4.242640687 Reorder the terms: -6 + x2 = -4.242640687 Solving -6 + x2 = -4.242640687 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '6' to each side of the equation. -6 + 6 + x2 = -4.242640687 + 6 Combine like terms: -6 + 6 = 0 0 + x2 = -4.242640687 + 6 x2 = -4.242640687 + 6 Combine like terms: -4.242640687 + 6 = 1.757359313 x2 = 1.757359313 Simplifying x2 = 1.757359313 Take the square root of each side: x = {-1.325654296, 1.325654296}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-3.200412581, 3.200412581, -1.325654296, 1.325654296}

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