0=x^4-109x^2+54

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Solution for 0=x^4-109x^2+54 equation:


Simplifying
0 = x4 + -109x2 + 54

Reorder the terms:
0 = 54 + -109x2 + x4

Solving
0 = 54 + -109x2 + x4

Solving for variable 'x'.

Combine like terms: 0 + -54 = -54
-54 + 109x2 + -1x4 = 54 + -109x2 + x4 + -54 + 109x2 + -1x4

Reorder the terms:
-54 + 109x2 + -1x4 = 54 + -54 + -109x2 + 109x2 + x4 + -1x4

Combine like terms: 54 + -54 = 0
-54 + 109x2 + -1x4 = 0 + -109x2 + 109x2 + x4 + -1x4
-54 + 109x2 + -1x4 = -109x2 + 109x2 + x4 + -1x4

Combine like terms: -109x2 + 109x2 = 0
-54 + 109x2 + -1x4 = 0 + x4 + -1x4
-54 + 109x2 + -1x4 = x4 + -1x4

Combine like terms: x4 + -1x4 = 0
-54 + 109x2 + -1x4 = 0

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

Divide each side by '-1'.
54 + -109x2 + x4 = 0

Move the constant term to the right:

Add '-54' to each side of the equation.
54 + -109x2 + -54 + x4 = 0 + -54

Reorder the terms:
54 + -54 + -109x2 + x4 = 0 + -54

Combine like terms: 54 + -54 = 0
0 + -109x2 + x4 = 0 + -54
-109x2 + x4 = 0 + -54

Combine like terms: 0 + -54 = -54
-109x2 + x4 = -54

The x term is -109x2.  Take half its coefficient (-54.5).
Square it (2970.25) and add it to both sides.

Add '2970.25' to each side of the equation.
-109x2 + 2970.25 + x4 = -54 + 2970.25

Reorder the terms:
2970.25 + -109x2 + x4 = -54 + 2970.25

Combine like terms: -54 + 2970.25 = 2916.25
2970.25 + -109x2 + x4 = 2916.25

Factor a perfect square on the left side:
(x2 + -54.5)(x2 + -54.5) = 2916.25

Calculate the square root of the right side: 54.002314765

Break this problem into two subproblems by setting 
(x2 + -54.5) equal to 54.002314765 and -54.002314765.

Subproblem 1

x2 + -54.5 = 54.002314765 Simplifying x2 + -54.5 = 54.002314765 Reorder the terms: -54.5 + x2 = 54.002314765 Solving -54.5 + x2 = 54.002314765 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '54.5' to each side of the equation. -54.5 + 54.5 + x2 = 54.002314765 + 54.5 Combine like terms: -54.5 + 54.5 = 0.0 0.0 + x2 = 54.002314765 + 54.5 x2 = 54.002314765 + 54.5 Combine like terms: 54.002314765 + 54.5 = 108.502314765 x2 = 108.502314765 Simplifying x2 = 108.502314765 Take the square root of each side: x = {-10.41644444, 10.41644444}

Subproblem 2

x2 + -54.5 = -54.002314765 Simplifying x2 + -54.5 = -54.002314765 Reorder the terms: -54.5 + x2 = -54.002314765 Solving -54.5 + x2 = -54.002314765 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '54.5' to each side of the equation. -54.5 + 54.5 + x2 = -54.002314765 + 54.5 Combine like terms: -54.5 + 54.5 = 0.0 0.0 + x2 = -54.002314765 + 54.5 x2 = -54.002314765 + 54.5 Combine like terms: -54.002314765 + 54.5 = 0.497685235 x2 = 0.497685235 Simplifying x2 = 0.497685235 Take the square root of each side: x = {-0.705468096, 0.705468096}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-10.41644444, 10.41644444, -0.705468096, 0.705468096}

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