6t+3(5t-4)=12t(2t-5)

Solution for 6t+3(5t-4)=12t(2t-5) equation:

Simplifying
6t + 3(5t + -4) = 12t(2t + -5)

Reorder the terms:
6t + 3(-4 + 5t) = 12t(2t + -5)
6t + (-4 * 3 + 5t * 3) = 12t(2t + -5)
6t + (-12 + 15t) = 12t(2t + -5)

Reorder the terms:
-12 + 6t + 15t = 12t(2t + -5)

Combine like terms: 6t + 15t = 21t
-12 + 21t = 12t(2t + -5)

Reorder the terms:
-12 + 21t = 12t(-5 + 2t)
-12 + 21t = (-5 * 12t + 2t * 12t)
-12 + 21t = (-60t + 24t2)

Solving
-12 + 21t = -60t + 24t2

Solving for variable 't'.

Combine like terms: 21t + 60t = 81t
-12 + 81t + -24t2 = -60t + 24t2 + 60t + -24t2

Reorder the terms:
-12 + 81t + -24t2 = -60t + 60t + 24t2 + -24t2

Combine like terms: -60t + 60t = 0
-12 + 81t + -24t2 = 0 + 24t2 + -24t2
-12 + 81t + -24t2 = 24t2 + -24t2

Combine like terms: 24t2 + -24t2 = 0
-12 + 81t + -24t2 = 0

Factor out the Greatest Common Factor (GCF), '3'.
3(-4 + 27t + -8t2) = 0

Ignore the factor 3.
Subproblem 1
Set the factor '(-4 + 27t + -8t2)' equal to zero and attempt to solve:

Simplifying
-4 + 27t + -8t2 = 0

Solving
-4 + 27t + -8t2 = 0

Begin completing the square.  Divide all terms by
-8 the coefficient of the squared term: 

Divide each side by '-8'.
0.5 + -3.375t + t2 = 0

Move the constant term to the right:

Add '-0.5' to each side of the equation.
0.5 + -3.375t + -0.5 + t2 = 0 + -0.5

Reorder the terms:
0.5 + -0.5 + -3.375t + t2 = 0 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + -3.375t + t2 = 0 + -0.5
-3.375t + t2 = 0 + -0.5

Combine like terms: 0 + -0.5 = -0.5
-3.375t + t2 = -0.5

The t term is -3.375t.  Take half its coefficient (-1.6875).
Square it (2.84765625) and add it to both sides.

Add '2.84765625' to each side of the equation.
-3.375t + 2.84765625 + t2 = -0.5 + 2.84765625

Reorder the terms:
2.84765625 + -3.375t + t2 = -0.5 + 2.84765625

Combine like terms: -0.5 + 2.84765625 = 2.34765625
2.84765625 + -3.375t + t2 = 2.34765625

Factor a perfect square on the left side:
(t + -1.6875)(t + -1.6875) = 2.34765625

Calculate the square root of the right side: 1.532206334

Break this problem into two subproblems by setting 
(t + -1.6875) equal to 1.532206334 and -1.532206334.

Subproblem 1
t + -1.6875 = 1.532206334

Simplifying
t + -1.6875 = 1.532206334

Reorder the terms:
-1.6875 + t = 1.532206334

Solving
-1.6875 + t = 1.532206334

Solving for variable 't'.

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

Add '1.6875' to each side of the equation.
-1.6875 + 1.6875 + t = 1.532206334 + 1.6875

Combine like terms: -1.6875 + 1.6875 = 0.0000
0.0000 + t = 1.532206334 + 1.6875
t = 1.532206334 + 1.6875

Combine like terms: 1.532206334 + 1.6875 = 3.219706334
t = 3.219706334

Simplifying
t = 3.219706334

Subproblem 2
t + -1.6875 = -1.532206334

Simplifying
t + -1.6875 = -1.532206334

Reorder the terms:
-1.6875 + t = -1.532206334

Solving
-1.6875 + t = -1.532206334

Solving for variable 't'.

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

Add '1.6875' to each side of the equation.
-1.6875 + 1.6875 + t = -1.532206334 + 1.6875

Combine like terms: -1.6875 + 1.6875 = 0.0000
0.0000 + t = -1.532206334 + 1.6875
t = -1.532206334 + 1.6875

Combine like terms: -1.532206334 + 1.6875 = 0.155293666
t = 0.155293666

Simplifying
t = 0.155293666

Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {3.219706334, 0.155293666}
Solution
t = {3.219706334, 0.155293666}