r^4-14r^2+33=0

Solution for r^4-14r^2+33=0 equation:

Simplifying
r4 + -14r2 + 33 = 0

Reorder the terms:
33 + -14r2 + r4 = 0

Solving
33 + -14r2 + r4 = 0

Solving for variable 'r'.

Factor a trinomial.
(3 + -1r2)(11 + -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 '(11 + -1r2)' equal to zero and attempt to solve:

Simplifying
11 + -1r2 = 0

Solving
11 + -1r2 = 0

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

Add '-11' to each side of the equation.
11 + -11 + -1r2 = 0 + -11

Combine like terms: 11 + -11 = 0
0 + -1r2 = 0 + -11
-1r2 = 0 + -11

Combine like terms: 0 + -11 = -11
-1r2 = -11

Divide each side by '-1'.
r2 = 11

Simplifying
r2 = 11

Take the square root of each side:
r = {-3.31662479, 3.31662479}
Solution
r = {-1.732050808, 1.732050808, -3.31662479, 3.31662479}