24x^4-54x^2+27=0

Solution for 24x^4-54x^2+27=0 equation:

Simplifying
24x4 + -54x2 + 27 = 0

Reorder the terms:
27 + -54x2 + 24x4 = 0

Solving
27 + -54x2 + 24x4 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '3'.
3(9 + -18x2 + 8x4) = 0

Factor a trinomial.
3((3 + -4x2)(3 + -2x2)) = 0

Ignore the factor 3.
Subproblem 1
Set the factor '(3 + -4x2)' equal to zero and attempt to solve:

Simplifying
3 + -4x2 = 0

Solving
3 + -4x2 = 0

Move all terms containing x to the left, all other terms to the right.

Add '-3' to each side of the equation.
3 + -3 + -4x2 = 0 + -3

Combine like terms: 3 + -3 = 0
0 + -4x2 = 0 + -3
-4x2 = 0 + -3

Combine like terms: 0 + -3 = -3
-4x2 = -3

Divide each side by '-4'.
x2 = 0.75

Simplifying
x2 = 0.75

Take the square root of each side:
x = {-0.866025404, 0.866025404}
Subproblem 2
Set the factor '(3 + -2x2)' equal to zero and attempt to solve:

Simplifying
3 + -2x2 = 0

Solving
3 + -2x2 = 0

Move all terms containing x to the left, all other terms to the right.

Add '-3' to each side of the equation.
3 + -3 + -2x2 = 0 + -3

Combine like terms: 3 + -3 = 0
0 + -2x2 = 0 + -3
-2x2 = 0 + -3

Combine like terms: 0 + -3 = -3
-2x2 = -3

Divide each side by '-2'.
x2 = 1.5

Simplifying
x2 = 1.5

Take the square root of each side:
x = {-1.224744871, 1.224744871}
Solution
x = {-0.866025404, 0.866025404, -1.224744871, 1.224744871}