x^4+-27x^2+121=0

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Solution for x^4+-27x^2+121=0 equation:


Simplifying
x4 + -27x2 + 121 = 0

Reorder the terms:
121 + -27x2 + x4 = 0

Solving
121 + -27x2 + x4 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-121' to each side of the equation.
121 + -27x2 + -121 + x4 = 0 + -121

Reorder the terms:
121 + -121 + -27x2 + x4 = 0 + -121

Combine like terms: 121 + -121 = 0
0 + -27x2 + x4 = 0 + -121
-27x2 + x4 = 0 + -121

Combine like terms: 0 + -121 = -121
-27x2 + x4 = -121

The x term is -27x2.  Take half its coefficient (-13.5).
Square it (182.25) and add it to both sides.

Add '182.25' to each side of the equation.
-27x2 + 182.25 + x4 = -121 + 182.25

Reorder the terms:
182.25 + -27x2 + x4 = -121 + 182.25

Combine like terms: -121 + 182.25 = 61.25
182.25 + -27x2 + x4 = 61.25

Factor a perfect square on the left side:
(x2 + -13.5)(x2 + -13.5) = 61.25

Calculate the square root of the right side: 7.826237921

Break this problem into two subproblems by setting 
(x2 + -13.5) equal to 7.826237921 and -7.826237921.

Subproblem 1

x2 + -13.5 = 7.826237921 Simplifying x2 + -13.5 = 7.826237921 Reorder the terms: -13.5 + x2 = 7.826237921 Solving -13.5 + x2 = 7.826237921 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '13.5' to each side of the equation. -13.5 + 13.5 + x2 = 7.826237921 + 13.5 Combine like terms: -13.5 + 13.5 = 0.0 0.0 + x2 = 7.826237921 + 13.5 x2 = 7.826237921 + 13.5 Combine like terms: 7.826237921 + 13.5 = 21.326237921 x2 = 21.326237921 Simplifying x2 = 21.326237921 Take the square root of each side: x = {-4.618033989, 4.618033989}

Subproblem 2

x2 + -13.5 = -7.826237921 Simplifying x2 + -13.5 = -7.826237921 Reorder the terms: -13.5 + x2 = -7.826237921 Solving -13.5 + x2 = -7.826237921 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '13.5' to each side of the equation. -13.5 + 13.5 + x2 = -7.826237921 + 13.5 Combine like terms: -13.5 + 13.5 = 0.0 0.0 + x2 = -7.826237921 + 13.5 x2 = -7.826237921 + 13.5 Combine like terms: -7.826237921 + 13.5 = 5.673762079 x2 = 5.673762079 Simplifying x2 = 5.673762079 Take the square root of each side: x = {-2.381966011, 2.381966011}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-4.618033989, 4.618033989, -2.381966011, 2.381966011}

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