6t(t-2)+9t=3(5t+3)-6

Solution for 6t(t-2)+9t=3(5t+3)-6 equation:

Simplifying
6t(t + -2) + 9t = 3(5t + 3) + -6

Reorder the terms:
6t(-2 + t) + 9t = 3(5t + 3) + -6
(-2 * 6t + t * 6t) + 9t = 3(5t + 3) + -6
(-12t + 6t2) + 9t = 3(5t + 3) + -6

Reorder the terms:
-12t + 9t + 6t2 = 3(5t + 3) + -6

Combine like terms: -12t + 9t = -3t
-3t + 6t2 = 3(5t + 3) + -6

Reorder the terms:
-3t + 6t2 = 3(3 + 5t) + -6
-3t + 6t2 = (3 * 3 + 5t * 3) + -6
-3t + 6t2 = (9 + 15t) + -6

Reorder the terms:
-3t + 6t2 = 9 + -6 + 15t

Combine like terms: 9 + -6 = 3
-3t + 6t2 = 3 + 15t

Solving
-3t + 6t2 = 3 + 15t

Solving for variable 't'.

Reorder the terms:
-3 + -3t + -15t + 6t2 = 3 + 15t + -3 + -15t

Combine like terms: -3t + -15t = -18t
-3 + -18t + 6t2 = 3 + 15t + -3 + -15t

Reorder the terms:
-3 + -18t + 6t2 = 3 + -3 + 15t + -15t

Combine like terms: 3 + -3 = 0
-3 + -18t + 6t2 = 0 + 15t + -15t
-3 + -18t + 6t2 = 15t + -15t

Combine like terms: 15t + -15t = 0
-3 + -18t + 6t2 = 0

Factor out the Greatest Common Factor (GCF), '3'.
3(-1 + -6t + 2t2) = 0

Ignore the factor 3.
Subproblem 1
Set the factor '(-1 + -6t + 2t2)' equal to zero and attempt to solve:

Simplifying
-1 + -6t + 2t2 = 0

Solving
-1 + -6t + 2t2 = 0

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

Divide each side by '2'.
-0.5 + -3t + t2 = 0

Move the constant term to the right:

Add '0.5' to each side of the equation.
-0.5 + -3t + 0.5 + t2 = 0 + 0.5

Reorder the terms:
-0.5 + 0.5 + -3t + t2 = 0 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + -3t + t2 = 0 + 0.5
-3t + t2 = 0 + 0.5

Combine like terms: 0 + 0.5 = 0.5
-3t + t2 = 0.5

The t term is -3t.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3t + 2.25 + t2 = 0.5 + 2.25

Reorder the terms:
2.25 + -3t + t2 = 0.5 + 2.25

Combine like terms: 0.5 + 2.25 = 2.75
2.25 + -3t + t2 = 2.75

Factor a perfect square on the left side:
(t + -1.5)(t + -1.5) = 2.75

Calculate the square root of the right side: 1.658312395

Break this problem into two subproblems by setting 
(t + -1.5) equal to 1.658312395 and -1.658312395.

Subproblem 1
t + -1.5 = 1.658312395

Simplifying
t + -1.5 = 1.658312395

Reorder the terms:
-1.5 + t = 1.658312395

Solving
-1.5 + t = 1.658312395

Solving for variable 't'.

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

Add '1.5' to each side of the equation.
-1.5 + 1.5 + t = 1.658312395 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + t = 1.658312395 + 1.5
t = 1.658312395 + 1.5

Combine like terms: 1.658312395 + 1.5 = 3.158312395
t = 3.158312395

Simplifying
t = 3.158312395

Subproblem 2
t + -1.5 = -1.658312395

Simplifying
t + -1.5 = -1.658312395

Reorder the terms:
-1.5 + t = -1.658312395

Solving
-1.5 + t = -1.658312395

Solving for variable 't'.

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

Add '1.5' to each side of the equation.
-1.5 + 1.5 + t = -1.658312395 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + t = -1.658312395 + 1.5
t = -1.658312395 + 1.5

Combine like terms: -1.658312395 + 1.5 = -0.158312395
t = -0.158312395

Simplifying
t = -0.158312395

Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {3.158312395, -0.158312395}
Solution
t = {3.158312395, -0.158312395}