x^4-13x^2+1=0

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Solution for x^4-13x^2+1=0 equation:


Simplifying
x4 + -13x2 + 1 = 0

Reorder the terms:
1 + -13x2 + x4 = 0

Solving
1 + -13x2 + 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 + -13x2 + -1 + x4 = 0 + -1

Reorder the terms:
1 + -1 + -13x2 + x4 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + -13x2 + x4 = 0 + -1
-13x2 + x4 = 0 + -1

Combine like terms: 0 + -1 = -1
-13x2 + x4 = -1

The x term is -13x2.  Take half its coefficient (-6.5).
Square it (42.25) and add it to both sides.

Add '42.25' to each side of the equation.
-13x2 + 42.25 + x4 = -1 + 42.25

Reorder the terms:
42.25 + -13x2 + x4 = -1 + 42.25

Combine like terms: -1 + 42.25 = 41.25
42.25 + -13x2 + x4 = 41.25

Factor a perfect square on the left side:
(x2 + -6.5)(x2 + -6.5) = 41.25

Calculate the square root of the right side: 6.422616289

Break this problem into two subproblems by setting 
(x2 + -6.5) equal to 6.422616289 and -6.422616289.

Subproblem 1

x2 + -6.5 = 6.422616289 Simplifying x2 + -6.5 = 6.422616289 Reorder the terms: -6.5 + x2 = 6.422616289 Solving -6.5 + x2 = 6.422616289 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '6.5' to each side of the equation. -6.5 + 6.5 + x2 = 6.422616289 + 6.5 Combine like terms: -6.5 + 6.5 = 0.0 0.0 + x2 = 6.422616289 + 6.5 x2 = 6.422616289 + 6.5 Combine like terms: 6.422616289 + 6.5 = 12.922616289 x2 = 12.922616289 Simplifying x2 = 12.922616289 Take the square root of each side: x = {-3.594804068, 3.594804068}

Subproblem 2

x2 + -6.5 = -6.422616289 Simplifying x2 + -6.5 = -6.422616289 Reorder the terms: -6.5 + x2 = -6.422616289 Solving -6.5 + x2 = -6.422616289 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '6.5' to each side of the equation. -6.5 + 6.5 + x2 = -6.422616289 + 6.5 Combine like terms: -6.5 + 6.5 = 0.0 0.0 + x2 = -6.422616289 + 6.5 x2 = -6.422616289 + 6.5 Combine like terms: -6.422616289 + 6.5 = 0.077383711 x2 = 0.077383711 Simplifying x2 = 0.077383711 Take the square root of each side: x = {-0.278179279, 0.278179279}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-3.594804068, 3.594804068, -0.278179279, 0.278179279}

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