2(t+3)t=3(t+2)

Solution for 2(t+3)t=3(t+2) equation:

Simplifying
2(t + 3) * t = 3(t + 2)

Reorder the terms:
2(3 + t) * t = 3(t + 2)

Reorder the terms for easier multiplication:
2t(3 + t) = 3(t + 2)
(3 * 2t + t * 2t) = 3(t + 2)
(6t + 2t2) = 3(t + 2)

Reorder the terms:
6t + 2t2 = 3(2 + t)
6t + 2t2 = (2 * 3 + t * 3)
6t + 2t2 = (6 + 3t)

Solving
6t + 2t2 = 6 + 3t

Solving for variable 't'.

Reorder the terms:
-6 + 6t + -3t + 2t2 = 6 + 3t + -6 + -3t

Combine like terms: 6t + -3t = 3t
-6 + 3t + 2t2 = 6 + 3t + -6 + -3t

Reorder the terms:
-6 + 3t + 2t2 = 6 + -6 + 3t + -3t

Combine like terms: 6 + -6 = 0
-6 + 3t + 2t2 = 0 + 3t + -3t
-6 + 3t + 2t2 = 3t + -3t

Combine like terms: 3t + -3t = 0
-6 + 3t + 2t2 = 0

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

Divide each side by '2'.
-3 + 1.5t + t2 = 0

Move the constant term to the right:

Add '3' to each side of the equation.
-3 + 1.5t + 3 + t2 = 0 + 3

Reorder the terms:
-3 + 3 + 1.5t + t2 = 0 + 3

Combine like terms: -3 + 3 = 0
0 + 1.5t + t2 = 0 + 3
1.5t + t2 = 0 + 3

Combine like terms: 0 + 3 = 3
1.5t + t2 = 3

The t term is 1.5t.  Take half its coefficient (0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
1.5t + 0.5625 + t2 = 3 + 0.5625

Reorder the terms:
0.5625 + 1.5t + t2 = 3 + 0.5625

Combine like terms: 3 + 0.5625 = 3.5625
0.5625 + 1.5t + t2 = 3.5625

Factor a perfect square on the left side:
(t + 0.75)(t + 0.75) = 3.5625

Calculate the square root of the right side: 1.887458609

Break this problem into two subproblems by setting 
(t + 0.75) equal to 1.887458609 and -1.887458609.

Subproblem 1
t + 0.75 = 1.887458609

Simplifying
t + 0.75 = 1.887458609

Reorder the terms:
0.75 + t = 1.887458609

Solving
0.75 + t = 1.887458609

Solving for variable 't'.

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

Add '-0.75' to each side of the equation.
0.75 + -0.75 + t = 1.887458609 + -0.75

Combine like terms: 0.75 + -0.75 = 0.00
0.00 + t = 1.887458609 + -0.75
t = 1.887458609 + -0.75

Combine like terms: 1.887458609 + -0.75 = 1.137458609
t = 1.137458609

Simplifying
t = 1.137458609

Subproblem 2
t + 0.75 = -1.887458609

Simplifying
t + 0.75 = -1.887458609

Reorder the terms:
0.75 + t = -1.887458609

Solving
0.75 + t = -1.887458609

Solving for variable 't'.

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

Add '-0.75' to each side of the equation.
0.75 + -0.75 + t = -1.887458609 + -0.75

Combine like terms: 0.75 + -0.75 = 0.00
0.00 + t = -1.887458609 + -0.75
t = -1.887458609 + -0.75

Combine like terms: -1.887458609 + -0.75 = -2.637458609
t = -2.637458609

Simplifying
t = -2.637458609

Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {1.137458609, -2.637458609}