x^4-11x^2+1=0

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Solution for x^4-11x^2+1=0 equation:


Simplifying
x4 + -11x2 + 1 = 0

Reorder the terms:
1 + -11x2 + x4 = 0

Solving
1 + -11x2 + x4 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + -11x2 + -1 + x4 = 0 + -1

Reorder the terms:
1 + -1 + -11x2 + x4 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + -11x2 + x4 = 0 + -1
-11x2 + x4 = 0 + -1

Combine like terms: 0 + -1 = -1
-11x2 + x4 = -1

The x term is -11x2.  Take half its coefficient (-5.5).
Square it (30.25) and add it to both sides.

Add '30.25' to each side of the equation.
-11x2 + 30.25 + x4 = -1 + 30.25

Reorder the terms:
30.25 + -11x2 + x4 = -1 + 30.25

Combine like terms: -1 + 30.25 = 29.25
30.25 + -11x2 + x4 = 29.25

Factor a perfect square on the left side:
(x2 + -5.5)(x2 + -5.5) = 29.25

Calculate the square root of the right side: 5.408326913

Break this problem into two subproblems by setting 
(x2 + -5.5) equal to 5.408326913 and -5.408326913.

Subproblem 1

x2 + -5.5 = 5.408326913 Simplifying x2 + -5.5 = 5.408326913 Reorder the terms: -5.5 + x2 = 5.408326913 Solving -5.5 + x2 = 5.408326913 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '5.5' to each side of the equation. -5.5 + 5.5 + x2 = 5.408326913 + 5.5 Combine like terms: -5.5 + 5.5 = 0.0 0.0 + x2 = 5.408326913 + 5.5 x2 = 5.408326913 + 5.5 Combine like terms: 5.408326913 + 5.5 = 10.908326913 x2 = 10.908326913 Simplifying x2 = 10.908326913 Take the square root of each side: x = {-3.302775638, 3.302775638}

Subproblem 2

x2 + -5.5 = -5.408326913 Simplifying x2 + -5.5 = -5.408326913 Reorder the terms: -5.5 + x2 = -5.408326913 Solving -5.5 + x2 = -5.408326913 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '5.5' to each side of the equation. -5.5 + 5.5 + x2 = -5.408326913 + 5.5 Combine like terms: -5.5 + 5.5 = 0.0 0.0 + x2 = -5.408326913 + 5.5 x2 = -5.408326913 + 5.5 Combine like terms: -5.408326913 + 5.5 = 0.091673087 x2 = 0.091673087 Simplifying x2 = 0.091673087 Take the square root of each side: x = {-0.302775638, 0.302775638}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-3.302775638, 3.302775638, -0.302775638, 0.302775638}

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