x^4-26x^2+1=0

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Solution for x^4-26x^2+1=0 equation:


Simplifying
x4 + -26x2 + 1 = 0

Reorder the terms:
1 + -26x2 + x4 = 0

Solving
1 + -26x2 + 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 + -26x2 + -1 + x4 = 0 + -1

Reorder the terms:
1 + -1 + -26x2 + x4 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + -26x2 + x4 = 0 + -1
-26x2 + x4 = 0 + -1

Combine like terms: 0 + -1 = -1
-26x2 + x4 = -1

The x term is -26x2.  Take half its coefficient (-13).
Square it (169) and add it to both sides.

Add '169' to each side of the equation.
-26x2 + 169 + x4 = -1 + 169

Reorder the terms:
169 + -26x2 + x4 = -1 + 169

Combine like terms: -1 + 169 = 168
169 + -26x2 + x4 = 168

Factor a perfect square on the left side:
(x2 + -13)(x2 + -13) = 168

Calculate the square root of the right side: 12.961481397

Break this problem into two subproblems by setting 
(x2 + -13) equal to 12.961481397 and -12.961481397.

Subproblem 1

x2 + -13 = 12.961481397 Simplifying x2 + -13 = 12.961481397 Reorder the terms: -13 + x2 = 12.961481397 Solving -13 + x2 = 12.961481397 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '13' to each side of the equation. -13 + 13 + x2 = 12.961481397 + 13 Combine like terms: -13 + 13 = 0 0 + x2 = 12.961481397 + 13 x2 = 12.961481397 + 13 Combine like terms: 12.961481397 + 13 = 25.961481397 x2 = 25.961481397 Simplifying x2 = 25.961481397 Take the square root of each side: x = {-5.095241054, 5.095241054}

Subproblem 2

x2 + -13 = -12.961481397 Simplifying x2 + -13 = -12.961481397 Reorder the terms: -13 + x2 = -12.961481397 Solving -13 + x2 = -12.961481397 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '13' to each side of the equation. -13 + 13 + x2 = -12.961481397 + 13 Combine like terms: -13 + 13 = 0 0 + x2 = -12.961481397 + 13 x2 = -12.961481397 + 13 Combine like terms: -12.961481397 + 13 = 0.038518603 x2 = 0.038518603 Simplifying x2 = 0.038518603 Take the square root of each side: x = {-0.196261568, 0.196261568}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-5.095241054, 5.095241054, -0.196261568, 0.196261568}

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