x^4-20x^2+92=0

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Solution for x^4-20x^2+92=0 equation:


Simplifying
x4 + -20x2 + 92 = 0

Reorder the terms:
92 + -20x2 + x4 = 0

Solving
92 + -20x2 + x4 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-92' to each side of the equation.
92 + -20x2 + -92 + x4 = 0 + -92

Reorder the terms:
92 + -92 + -20x2 + x4 = 0 + -92

Combine like terms: 92 + -92 = 0
0 + -20x2 + x4 = 0 + -92
-20x2 + x4 = 0 + -92

Combine like terms: 0 + -92 = -92
-20x2 + x4 = -92

The x term is -20x2.  Take half its coefficient (-10).
Square it (100) and add it to both sides.

Add '100' to each side of the equation.
-20x2 + 100 + x4 = -92 + 100

Reorder the terms:
100 + -20x2 + x4 = -92 + 100

Combine like terms: -92 + 100 = 8
100 + -20x2 + x4 = 8

Factor a perfect square on the left side:
(x2 + -10)(x2 + -10) = 8

Calculate the square root of the right side: 2.828427125

Break this problem into two subproblems by setting 
(x2 + -10) equal to 2.828427125 and -2.828427125.

Subproblem 1

x2 + -10 = 2.828427125 Simplifying x2 + -10 = 2.828427125 Reorder the terms: -10 + x2 = 2.828427125 Solving -10 + x2 = 2.828427125 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '10' to each side of the equation. -10 + 10 + x2 = 2.828427125 + 10 Combine like terms: -10 + 10 = 0 0 + x2 = 2.828427125 + 10 x2 = 2.828427125 + 10 Combine like terms: 2.828427125 + 10 = 12.828427125 x2 = 12.828427125 Simplifying x2 = 12.828427125 Take the square root of each side: x = {-3.581679372, 3.581679372}

Subproblem 2

x2 + -10 = -2.828427125 Simplifying x2 + -10 = -2.828427125 Reorder the terms: -10 + x2 = -2.828427125 Solving -10 + x2 = -2.828427125 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '10' to each side of the equation. -10 + 10 + x2 = -2.828427125 + 10 Combine like terms: -10 + 10 = 0 0 + x2 = -2.828427125 + 10 x2 = -2.828427125 + 10 Combine like terms: -2.828427125 + 10 = 7.171572875 x2 = 7.171572875 Simplifying x2 = 7.171572875 Take the square root of each side: x = {-2.677979252, 2.677979252}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-3.581679372, 3.581679372, -2.677979252, 2.677979252}

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