4t^2+6rt+2r^2=0

Solution for 4t^2+6rt+2r^2=0 equation:

Simplifying
4t2 + 6rt + 2r2 = 0

Reorder the terms:
6rt + 2r2 + 4t2 = 0

Solving
6rt + 2r2 + 4t2 = 0

Solving for variable 'r'.

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

Factor a trinomial.
2((r + t)(r + 2t)) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(r + t)' equal to zero and attempt to solve:

Simplifying
r + t = 0

Solving
r + t = 0

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

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

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

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

Simplifying
r + 2t = 0

Solving
r + 2t = 0

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

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

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

Simplifying
r = -2t
Solution
r = {-1t, -2t}