x^4-13x^2+38=0

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Solution for x^4-13x^2+38=0 equation:


Simplifying
x4 + -13x2 + 38 = 0

Reorder the terms:
38 + -13x2 + x4 = 0

Solving
38 + -13x2 + x4 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-38' to each side of the equation.
38 + -13x2 + -38 + x4 = 0 + -38

Reorder the terms:
38 + -38 + -13x2 + x4 = 0 + -38

Combine like terms: 38 + -38 = 0
0 + -13x2 + x4 = 0 + -38
-13x2 + x4 = 0 + -38

Combine like terms: 0 + -38 = -38
-13x2 + x4 = -38

The x term is -13x2.  Take half its coefficient (-6.5).
Square it (42.25) and add it to both sides.

Add '42.25' to each side of the equation.
-13x2 + 42.25 + x4 = -38 + 42.25

Reorder the terms:
42.25 + -13x2 + x4 = -38 + 42.25

Combine like terms: -38 + 42.25 = 4.25
42.25 + -13x2 + x4 = 4.25

Factor a perfect square on the left side:
(x2 + -6.5)(x2 + -6.5) = 4.25

Calculate the square root of the right side: 2.061552813

Break this problem into two subproblems by setting 
(x2 + -6.5) equal to 2.061552813 and -2.061552813.

Subproblem 1

x2 + -6.5 = 2.061552813 Simplifying x2 + -6.5 = 2.061552813 Reorder the terms: -6.5 + x2 = 2.061552813 Solving -6.5 + x2 = 2.061552813 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '6.5' to each side of the equation. -6.5 + 6.5 + x2 = 2.061552813 + 6.5 Combine like terms: -6.5 + 6.5 = 0.0 0.0 + x2 = 2.061552813 + 6.5 x2 = 2.061552813 + 6.5 Combine like terms: 2.061552813 + 6.5 = 8.561552813 x2 = 8.561552813 Simplifying x2 = 8.561552813 Take the square root of each side: x = {-2.926013126, 2.926013126}

Subproblem 2

x2 + -6.5 = -2.061552813 Simplifying x2 + -6.5 = -2.061552813 Reorder the terms: -6.5 + x2 = -2.061552813 Solving -6.5 + x2 = -2.061552813 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '6.5' to each side of the equation. -6.5 + 6.5 + x2 = -2.061552813 + 6.5 Combine like terms: -6.5 + 6.5 = 0.0 0.0 + x2 = -2.061552813 + 6.5 x2 = -2.061552813 + 6.5 Combine like terms: -2.061552813 + 6.5 = 4.438447187 x2 = 4.438447187 Simplifying x2 = 4.438447187 Take the square root of each side: x = {-2.106762252, 2.106762252}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-2.926013126, 2.926013126, -2.106762252, 2.106762252}

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