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

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


Simplifying
4x4 + -13x2 + 1 = 0

Reorder the terms:
1 + -13x2 + 4x4 = 0

Solving
1 + -13x2 + 4x4 = 0

Solving for variable 'x'.

Begin completing the square.  Divide all terms by
4 the coefficient of the squared term: 

Divide each side by '4'.
0.25 + -3.25x2 + x4 = 0

Move the constant term to the right:

Add '-0.25' to each side of the equation.
0.25 + -3.25x2 + -0.25 + x4 = 0 + -0.25

Reorder the terms:
0.25 + -0.25 + -3.25x2 + x4 = 0 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + -3.25x2 + x4 = 0 + -0.25
-3.25x2 + x4 = 0 + -0.25

Combine like terms: 0 + -0.25 = -0.25
-3.25x2 + x4 = -0.25

The x term is -3.25x2.  Take half its coefficient (-1.625).
Square it (2.640625) and add it to both sides.

Add '2.640625' to each side of the equation.
-3.25x2 + 2.640625 + x4 = -0.25 + 2.640625

Reorder the terms:
2.640625 + -3.25x2 + x4 = -0.25 + 2.640625

Combine like terms: -0.25 + 2.640625 = 2.390625
2.640625 + -3.25x2 + x4 = 2.390625

Factor a perfect square on the left side:
(x2 + -1.625)(x2 + -1.625) = 2.390625

Calculate the square root of the right side: 1.54616461

Break this problem into two subproblems by setting 
(x2 + -1.625) equal to 1.54616461 and -1.54616461.

Subproblem 1

x2 + -1.625 = 1.54616461 Simplifying x2 + -1.625 = 1.54616461 Reorder the terms: -1.625 + x2 = 1.54616461 Solving -1.625 + x2 = 1.54616461 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.625' to each side of the equation. -1.625 + 1.625 + x2 = 1.54616461 + 1.625 Combine like terms: -1.625 + 1.625 = 0.000 0.000 + x2 = 1.54616461 + 1.625 x2 = 1.54616461 + 1.625 Combine like terms: 1.54616461 + 1.625 = 3.17116461 x2 = 3.17116461 Simplifying x2 = 3.17116461 Take the square root of each side: x = {-1.780776407, 1.780776407}

Subproblem 2

x2 + -1.625 = -1.54616461 Simplifying x2 + -1.625 = -1.54616461 Reorder the terms: -1.625 + x2 = -1.54616461 Solving -1.625 + x2 = -1.54616461 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.625' to each side of the equation. -1.625 + 1.625 + x2 = -1.54616461 + 1.625 Combine like terms: -1.625 + 1.625 = 0.000 0.000 + x2 = -1.54616461 + 1.625 x2 = -1.54616461 + 1.625 Combine like terms: -1.54616461 + 1.625 = 0.07883539 x2 = 0.07883539 Simplifying x2 = 0.07883539 Take the square root of each side: x = {-0.280776406, 0.280776406}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-1.780776407, 1.780776407, -0.280776406, 0.280776406}

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