r^4-9r^2+18=0

Solution for r^4-9r^2+18=0 equation:

Simplifying
r4 + -9r2 + 18 = 0

Reorder the terms:
18 + -9r2 + r4 = 0

Solving
18 + -9r2 + r4 = 0

Solving for variable 'r'.

Factor a trinomial.
(3 + -1r2)(6 + -1r2) = 0

Subproblem 1
Set the factor '(3 + -1r2)' equal to zero and attempt to solve:

Simplifying
3 + -1r2 = 0

Solving
3 + -1r2 = 0

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

Add '-3' to each side of the equation.
3 + -3 + -1r2 = 0 + -3

Combine like terms: 3 + -3 = 0
0 + -1r2 = 0 + -3
-1r2 = 0 + -3

Combine like terms: 0 + -3 = -3
-1r2 = -3

Divide each side by '-1'.
r2 = 3

Simplifying
r2 = 3

Take the square root of each side:
r = {-1.732050808, 1.732050808}
Subproblem 2
Set the factor '(6 + -1r2)' equal to zero and attempt to solve:

Simplifying
6 + -1r2 = 0

Solving
6 + -1r2 = 0

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

Add '-6' to each side of the equation.
6 + -6 + -1r2 = 0 + -6

Combine like terms: 6 + -6 = 0
0 + -1r2 = 0 + -6
-1r2 = 0 + -6

Combine like terms: 0 + -6 = -6
-1r2 = -6

Divide each side by '-1'.
r2 = 6

Simplifying
r2 = 6

Take the square root of each side:
r = {-2.449489743, 2.449489743}
Solution
r = {-1.732050808, 1.732050808, -2.449489743, 2.449489743}