x^4-16x^2+24=0

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Solution for x^4-16x^2+24=0 equation:


Simplifying
x4 + -16x2 + 24 = 0

Reorder the terms:
24 + -16x2 + x4 = 0

Solving
24 + -16x2 + x4 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-24' to each side of the equation.
24 + -16x2 + -24 + x4 = 0 + -24

Reorder the terms:
24 + -24 + -16x2 + x4 = 0 + -24

Combine like terms: 24 + -24 = 0
0 + -16x2 + x4 = 0 + -24
-16x2 + x4 = 0 + -24

Combine like terms: 0 + -24 = -24
-16x2 + x4 = -24

The x term is -16x2.  Take half its coefficient (-8).
Square it (64) and add it to both sides.

Add '64' to each side of the equation.
-16x2 + 64 + x4 = -24 + 64

Reorder the terms:
64 + -16x2 + x4 = -24 + 64

Combine like terms: -24 + 64 = 40
64 + -16x2 + x4 = 40

Factor a perfect square on the left side:
(x2 + -8)(x2 + -8) = 40

Calculate the square root of the right side: 6.32455532

Break this problem into two subproblems by setting 
(x2 + -8) equal to 6.32455532 and -6.32455532.

Subproblem 1

x2 + -8 = 6.32455532 Simplifying x2 + -8 = 6.32455532 Reorder the terms: -8 + x2 = 6.32455532 Solving -8 + x2 = 6.32455532 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '8' to each side of the equation. -8 + 8 + x2 = 6.32455532 + 8 Combine like terms: -8 + 8 = 0 0 + x2 = 6.32455532 + 8 x2 = 6.32455532 + 8 Combine like terms: 6.32455532 + 8 = 14.32455532 x2 = 14.32455532 Simplifying x2 = 14.32455532 Take the square root of each side: x = {-3.784779428, 3.784779428}

Subproblem 2

x2 + -8 = -6.32455532 Simplifying x2 + -8 = -6.32455532 Reorder the terms: -8 + x2 = -6.32455532 Solving -8 + x2 = -6.32455532 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '8' to each side of the equation. -8 + 8 + x2 = -6.32455532 + 8 Combine like terms: -8 + 8 = 0 0 + x2 = -6.32455532 + 8 x2 = -6.32455532 + 8 Combine like terms: -6.32455532 + 8 = 1.67544468 x2 = 1.67544468 Simplifying x2 = 1.67544468 Take the square root of each side: x = {-1.294389694, 1.294389694}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-3.784779428, 3.784779428, -1.294389694, 1.294389694}

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