x^4-24x^2+20=0

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Solution for x^4-24x^2+20=0 equation:


Simplifying
x4 + -24x2 + 20 = 0

Reorder the terms:
20 + -24x2 + x4 = 0

Solving
20 + -24x2 + x4 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-20' to each side of the equation.
20 + -24x2 + -20 + x4 = 0 + -20

Reorder the terms:
20 + -20 + -24x2 + x4 = 0 + -20

Combine like terms: 20 + -20 = 0
0 + -24x2 + x4 = 0 + -20
-24x2 + x4 = 0 + -20

Combine like terms: 0 + -20 = -20
-24x2 + x4 = -20

The x term is -24x2.  Take half its coefficient (-12).
Square it (144) and add it to both sides.

Add '144' to each side of the equation.
-24x2 + 144 + x4 = -20 + 144

Reorder the terms:
144 + -24x2 + x4 = -20 + 144

Combine like terms: -20 + 144 = 124
144 + -24x2 + x4 = 124

Factor a perfect square on the left side:
(x2 + -12)(x2 + -12) = 124

Calculate the square root of the right side: 11.135528726

Break this problem into two subproblems by setting 
(x2 + -12) equal to 11.135528726 and -11.135528726.

Subproblem 1

x2 + -12 = 11.135528726 Simplifying x2 + -12 = 11.135528726 Reorder the terms: -12 + x2 = 11.135528726 Solving -12 + x2 = 11.135528726 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '12' to each side of the equation. -12 + 12 + x2 = 11.135528726 + 12 Combine like terms: -12 + 12 = 0 0 + x2 = 11.135528726 + 12 x2 = 11.135528726 + 12 Combine like terms: 11.135528726 + 12 = 23.135528726 x2 = 23.135528726 Simplifying x2 = 23.135528726 Take the square root of each side: x = {-4.809940616, 4.809940616}

Subproblem 2

x2 + -12 = -11.135528726 Simplifying x2 + -12 = -11.135528726 Reorder the terms: -12 + x2 = -11.135528726 Solving -12 + x2 = -11.135528726 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '12' to each side of the equation. -12 + 12 + x2 = -11.135528726 + 12 Combine like terms: -12 + 12 = 0 0 + x2 = -11.135528726 + 12 x2 = -11.135528726 + 12 Combine like terms: -11.135528726 + 12 = 0.864471274 x2 = 0.864471274 Simplifying x2 = 0.864471274 Take the square root of each side: x = {-0.929769474, 0.929769474}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-4.809940616, 4.809940616, -0.929769474, 0.929769474}

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