x^4-14x^2+1=0

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Solution for x^4-14x^2+1=0 equation:


Simplifying
x4 + -14x2 + 1 = 0

Reorder the terms:
1 + -14x2 + x4 = 0

Solving
1 + -14x2 + 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 + -14x2 + -1 + x4 = 0 + -1

Reorder the terms:
1 + -1 + -14x2 + x4 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + -14x2 + x4 = 0 + -1
-14x2 + x4 = 0 + -1

Combine like terms: 0 + -1 = -1
-14x2 + x4 = -1

The x term is -14x2.  Take half its coefficient (-7).
Square it (49) and add it to both sides.

Add '49' to each side of the equation.
-14x2 + 49 + x4 = -1 + 49

Reorder the terms:
49 + -14x2 + x4 = -1 + 49

Combine like terms: -1 + 49 = 48
49 + -14x2 + x4 = 48

Factor a perfect square on the left side:
(x2 + -7)(x2 + -7) = 48

Calculate the square root of the right side: 6.92820323

Break this problem into two subproblems by setting 
(x2 + -7) equal to 6.92820323 and -6.92820323.

Subproblem 1

x2 + -7 = 6.92820323 Simplifying x2 + -7 = 6.92820323 Reorder the terms: -7 + x2 = 6.92820323 Solving -7 + x2 = 6.92820323 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '7' to each side of the equation. -7 + 7 + x2 = 6.92820323 + 7 Combine like terms: -7 + 7 = 0 0 + x2 = 6.92820323 + 7 x2 = 6.92820323 + 7 Combine like terms: 6.92820323 + 7 = 13.92820323 x2 = 13.92820323 Simplifying x2 = 13.92820323 Take the square root of each side: x = {-3.732050808, 3.732050808}

Subproblem 2

x2 + -7 = -6.92820323 Simplifying x2 + -7 = -6.92820323 Reorder the terms: -7 + x2 = -6.92820323 Solving -7 + x2 = -6.92820323 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '7' to each side of the equation. -7 + 7 + x2 = -6.92820323 + 7 Combine like terms: -7 + 7 = 0 0 + x2 = -6.92820323 + 7 x2 = -6.92820323 + 7 Combine like terms: -6.92820323 + 7 = 0.07179677 x2 = 0.07179677 Simplifying x2 = 0.07179677 Take the square root of each side: x = {-0.267949193, 0.267949193}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-3.732050808, 3.732050808, -0.267949193, 0.267949193}

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