20r^4-39r^2+18=0

Solution for 20r^4-39r^2+18=0 equation:

Simplifying
20r4 + -39r2 + 18 = 0

Reorder the terms:
18 + -39r2 + 20r4 = 0

Solving
18 + -39r2 + 20r4 = 0

Solving for variable 'r'.

Factor a trinomial.
(3 + -4r2)(6 + -5r2) = 0

Subproblem 1
Set the factor '(3 + -4r2)' equal to zero and attempt to solve:

Simplifying
3 + -4r2 = 0

Solving
3 + -4r2 = 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 + -4r2 = 0 + -3

Combine like terms: 3 + -3 = 0
0 + -4r2 = 0 + -3
-4r2 = 0 + -3

Combine like terms: 0 + -3 = -3
-4r2 = -3

Divide each side by '-4'.
r2 = 0.75

Simplifying
r2 = 0.75

Take the square root of each side:
r = {-0.866025404, 0.866025404}
Subproblem 2
Set the factor '(6 + -5r2)' equal to zero and attempt to solve:

Simplifying
6 + -5r2 = 0

Solving
6 + -5r2 = 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 + -5r2 = 0 + -6

Combine like terms: 6 + -6 = 0
0 + -5r2 = 0 + -6
-5r2 = 0 + -6

Combine like terms: 0 + -6 = -6
-5r2 = -6

Divide each side by '-5'.
r2 = 1.2

Simplifying
r2 = 1.2

Take the square root of each side:
r = {-1.095445115, 1.095445115}
Solution
r = {-0.866025404, 0.866025404, -1.095445115, 1.095445115}