x^4-18x^2+36=0

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Solution for x^4-18x^2+36=0 equation:


Simplifying
x4 + -18x2 + 36 = 0

Reorder the terms:
36 + -18x2 + x4 = 0

Solving
36 + -18x2 + x4 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-36' to each side of the equation.
36 + -18x2 + -36 + x4 = 0 + -36

Reorder the terms:
36 + -36 + -18x2 + x4 = 0 + -36

Combine like terms: 36 + -36 = 0
0 + -18x2 + x4 = 0 + -36
-18x2 + x4 = 0 + -36

Combine like terms: 0 + -36 = -36
-18x2 + x4 = -36

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 = -36 + 81

Reorder the terms:
81 + -18x2 + x4 = -36 + 81

Combine like terms: -36 + 81 = 45
81 + -18x2 + x4 = 45

Factor a perfect square on the left side:
(x2 + -9)(x2 + -9) = 45

Calculate the square root of the right side: 6.708203933

Break this problem into two subproblems by setting 
(x2 + -9) equal to 6.708203933 and -6.708203933.

Subproblem 1

x2 + -9 = 6.708203933 Simplifying x2 + -9 = 6.708203933 Reorder the terms: -9 + x2 = 6.708203933 Solving -9 + x2 = 6.708203933 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 = 6.708203933 + 9 Combine like terms: -9 + 9 = 0 0 + x2 = 6.708203933 + 9 x2 = 6.708203933 + 9 Combine like terms: 6.708203933 + 9 = 15.708203933 x2 = 15.708203933 Simplifying x2 = 15.708203933 Take the square root of each side: x = {-3.963357659, 3.963357659}

Subproblem 2

x2 + -9 = -6.708203933 Simplifying x2 + -9 = -6.708203933 Reorder the terms: -9 + x2 = -6.708203933 Solving -9 + x2 = -6.708203933 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 = -6.708203933 + 9 Combine like terms: -9 + 9 = 0 0 + x2 = -6.708203933 + 9 x2 = -6.708203933 + 9 Combine like terms: -6.708203933 + 9 = 2.291796067 x2 = 2.291796067 Simplifying x2 = 2.291796067 Take the square root of each side: x = {-1.513867916, 1.513867916}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-3.963357659, 3.963357659, -1.513867916, 1.513867916}

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