a^4=6a^2+27

Solution for a^4=6a^2+27 equation:

Simplifying
a4 = 6a2 + 27

Reorder the terms:
a4 = 27 + 6a2

Solving
a4 = 27 + 6a2

Solving for variable 'a'.

Reorder the terms:
-27 + -6a2 + a4 = 27 + 6a2 + -27 + -6a2

Reorder the terms:
-27 + -6a2 + a4 = 27 + -27 + 6a2 + -6a2

Combine like terms: 27 + -27 = 0
-27 + -6a2 + a4 = 0 + 6a2 + -6a2
-27 + -6a2 + a4 = 6a2 + -6a2

Combine like terms: 6a2 + -6a2 = 0
-27 + -6a2 + a4 = 0

Factor a trinomial.
(-3 + -1a2)(9 + -1a2) = 0

Factor a difference between two squares.
(-3 + -1a2)((3 + a)(3 + -1a)) = 0

Subproblem 1
Set the factor '(-3 + -1a2)' equal to zero and attempt to solve:

Simplifying
-3 + -1a2 = 0

Solving
-3 + -1a2 = 0

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

Add '3' to each side of the equation.
-3 + 3 + -1a2 = 0 + 3

Combine like terms: -3 + 3 = 0
0 + -1a2 = 0 + 3
-1a2 = 0 + 3

Combine like terms: 0 + 3 = 3
-1a2 = 3

Divide each side by '-1'.
a2 = -3

Simplifying
a2 = -3

The solution to this equation could not be determined.

This subproblem is being ignored because a solution could not be determined.
Subproblem 2
Set the factor '(3 + a)' equal to zero and attempt to solve:

Simplifying
3 + a = 0

Solving
3 + a = 0

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

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

Combine like terms: 3 + -3 = 0
0 + a = 0 + -3
a = 0 + -3

Combine like terms: 0 + -3 = -3
a = -3

Simplifying
a = -3
Subproblem 3
Set the factor '(3 + -1a)' equal to zero and attempt to solve:

Simplifying
3 + -1a = 0

Solving
3 + -1a = 0

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

Add '-3' to each side of the equation.
3 + -3 + -1a = 0 + -3

Combine like terms: 3 + -3 = 0
0 + -1a = 0 + -3
-1a = 0 + -3

Combine like terms: 0 + -3 = -3
-1a = -3

Divide each side by '-1'.
a = 3

Simplifying
a = 3
Solution
a = {-3, 3}