3r^3+6r^2t-24rt^2=0

Solution for 3r^3+6r^2t-24rt^2=0 equation:

Simplifying
3r3 + 6r2t + -24rt2 = 0

Reorder the terms:
-24rt2 + 6r2t + 3r3 = 0

Solving
-24rt2 + 6r2t + 3r3 = 0

Solving for variable 'r'.

Factor out the Greatest Common Factor (GCF), '3r'.
3r(-8t2 + 2rt + r2) = 0

Factor a trinomial.
3r((-4t + -1r)(2t + -1r)) = 0

Ignore the factor 3.
Subproblem 1
Set the factor 'r' equal to zero and attempt to solve:

Simplifying
r = 0

Solving
r = 0

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

Simplifying
r = 0
Subproblem 2
Set the factor '(-4t + -1r)' equal to zero and attempt to solve:

Simplifying
-4t + -1r = 0

Reorder the terms:
-1r + -4t = 0

Solving
-1r + -4t = 0

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

Add '4t' to each side of the equation.
-1r + -4t + 4t = 0 + 4t

Combine like terms: -4t + 4t = 0
-1r + 0 = 0 + 4t
-1r = 0 + 4t
Remove the zero:
-1r = 4t

Divide each side by '-1'.
r = -4t

Simplifying
r = -4t
Subproblem 3
Set the factor '(2t + -1r)' equal to zero and attempt to solve:

Simplifying
2t + -1r = 0

Reorder the terms:
-1r + 2t = 0

Solving
-1r + 2t = 0

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

Add '-2t' to each side of the equation.
-1r + 2t + -2t = 0 + -2t

Combine like terms: 2t + -2t = 0
-1r + 0 = 0 + -2t
-1r = 0 + -2t
Remove the zero:
-1r = -2t

Divide each side by '-1'.
r = 2t

Simplifying
r = 2t
Solution
r = {0, -4t, 2t}