x^4-8x^2+13=0

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Solution for x^4-8x^2+13=0 equation:


Simplifying
x4 + -8x2 + 13 = 0

Reorder the terms:
13 + -8x2 + x4 = 0

Solving
13 + -8x2 + x4 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-13' to each side of the equation.
13 + -8x2 + -13 + x4 = 0 + -13

Reorder the terms:
13 + -13 + -8x2 + x4 = 0 + -13

Combine like terms: 13 + -13 = 0
0 + -8x2 + x4 = 0 + -13
-8x2 + x4 = 0 + -13

Combine like terms: 0 + -13 = -13
-8x2 + x4 = -13

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 = -13 + 16

Reorder the terms:
16 + -8x2 + x4 = -13 + 16

Combine like terms: -13 + 16 = 3
16 + -8x2 + x4 = 3

Factor a perfect square on the left side:
(x2 + -4)(x2 + -4) = 3

Calculate the square root of the right side: 1.732050808

Break this problem into two subproblems by setting 
(x2 + -4) equal to 1.732050808 and -1.732050808.

Subproblem 1

x2 + -4 = 1.732050808 Simplifying x2 + -4 = 1.732050808 Reorder the terms: -4 + x2 = 1.732050808 Solving -4 + x2 = 1.732050808 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 = 1.732050808 + 4 Combine like terms: -4 + 4 = 0 0 + x2 = 1.732050808 + 4 x2 = 1.732050808 + 4 Combine like terms: 1.732050808 + 4 = 5.732050808 x2 = 5.732050808 Simplifying x2 = 5.732050808 Take the square root of each side: x = {-2.394170171, 2.394170171}

Subproblem 2

x2 + -4 = -1.732050808 Simplifying x2 + -4 = -1.732050808 Reorder the terms: -4 + x2 = -1.732050808 Solving -4 + x2 = -1.732050808 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 = -1.732050808 + 4 Combine like terms: -4 + 4 = 0 0 + x2 = -1.732050808 + 4 x2 = -1.732050808 + 4 Combine like terms: -1.732050808 + 4 = 2.267949192 x2 = 2.267949192 Simplifying x2 = 2.267949192 Take the square root of each side: x = {-1.505971179, 1.505971179}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-2.394170171, 2.394170171, -1.505971179, 1.505971179}

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