4x^4-32x^2+13=0

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Solution for 4x^4-32x^2+13=0 equation:


Simplifying
4x4 + -32x2 + 13 = 0

Reorder the terms:
13 + -32x2 + 4x4 = 0

Solving
13 + -32x2 + 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'.
3.25 + -8x2 + x4 = 0

Move the constant term to the right:

Add '-3.25' to each side of the equation.
3.25 + -8x2 + -3.25 + x4 = 0 + -3.25

Reorder the terms:
3.25 + -3.25 + -8x2 + x4 = 0 + -3.25

Combine like terms: 3.25 + -3.25 = 0.00
0.00 + -8x2 + x4 = 0 + -3.25
-8x2 + x4 = 0 + -3.25

Combine like terms: 0 + -3.25 = -3.25
-8x2 + x4 = -3.25

The x term is -8x2.  Take half its coefficient (-4).
Square it (16) and add it to both sides.

Add '16' to each side of the equation.
-8x2 + 16 + x4 = -3.25 + 16

Reorder the terms:
16 + -8x2 + x4 = -3.25 + 16

Combine like terms: -3.25 + 16 = 12.75
16 + -8x2 + x4 = 12.75

Factor a perfect square on the left side:
(x2 + -4)(x2 + -4) = 12.75

Calculate the square root of the right side: 3.570714214

Break this problem into two subproblems by setting 
(x2 + -4) equal to 3.570714214 and -3.570714214.

Subproblem 1

x2 + -4 = 3.570714214 Simplifying x2 + -4 = 3.570714214 Reorder the terms: -4 + x2 = 3.570714214 Solving -4 + x2 = 3.570714214 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + x2 = 3.570714214 + 4 Combine like terms: -4 + 4 = 0 0 + x2 = 3.570714214 + 4 x2 = 3.570714214 + 4 Combine like terms: 3.570714214 + 4 = 7.570714214 x2 = 7.570714214 Simplifying x2 = 7.570714214 Take the square root of each side: x = {-2.751493088, 2.751493088}

Subproblem 2

x2 + -4 = -3.570714214 Simplifying x2 + -4 = -3.570714214 Reorder the terms: -4 + x2 = -3.570714214 Solving -4 + x2 = -3.570714214 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + x2 = -3.570714214 + 4 Combine like terms: -4 + 4 = 0 0 + x2 = -3.570714214 + 4 x2 = -3.570714214 + 4 Combine like terms: -3.570714214 + 4 = 0.429285786 x2 = 0.429285786 Simplifying x2 = 0.429285786 Take the square root of each side: x = {-0.655199043, 0.655199043}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-2.751493088, 2.751493088, -0.655199043, 0.655199043}

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