2(2t+3)*2t=48

Solution for 2(2t+3)*2t=48 equation:

Simplifying
2(2t + 3) * 2t = 48

Reorder the terms:
2(3 + 2t) * 2t = 48

Reorder the terms for easier multiplication:
2 * 2t(3 + 2t) = 48

Multiply 2 * 2
4t(3 + 2t) = 48
(3 * 4t + 2t * 4t) = 48
(12t + 8t2) = 48

Solving
12t + 8t2 = 48

Solving for variable 't'.

Reorder the terms:
-48 + 12t + 8t2 = 48 + -48

Combine like terms: 48 + -48 = 0
-48 + 12t + 8t2 = 0

Factor out the Greatest Common Factor (GCF), '4'.
4(-12 + 3t + 2t2) = 0

Ignore the factor 4.
Subproblem 1
Set the factor '(-12 + 3t + 2t2)' equal to zero and attempt to solve:

Simplifying
-12 + 3t + 2t2 = 0

Solving
-12 + 3t + 2t2 = 0

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

Divide each side by '2'.
-6 + 1.5t + t2 = 0

Move the constant term to the right:

Add '6' to each side of the equation.
-6 + 1.5t + 6 + t2 = 0 + 6

Reorder the terms:
-6 + 6 + 1.5t + t2 = 0 + 6

Combine like terms: -6 + 6 = 0
0 + 1.5t + t2 = 0 + 6
1.5t + t2 = 0 + 6

Combine like terms: 0 + 6 = 6
1.5t + t2 = 6

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 = 6 + 0.5625

Reorder the terms:
0.5625 + 1.5t + t2 = 6 + 0.5625

Combine like terms: 6 + 0.5625 = 6.5625
0.5625 + 1.5t + t2 = 6.5625

Factor a perfect square on the left side:
(t + 0.75)(t + 0.75) = 6.5625

Calculate the square root of the right side: 2.561737691

Break this problem into two subproblems by setting 
(t + 0.75) equal to 2.561737691 and -2.561737691.

Subproblem 1
t + 0.75 = 2.561737691

Simplifying
t + 0.75 = 2.561737691

Reorder the terms:
0.75 + t = 2.561737691

Solving
0.75 + t = 2.561737691

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 = 2.561737691 + -0.75

Combine like terms: 0.75 + -0.75 = 0.00
0.00 + t = 2.561737691 + -0.75
t = 2.561737691 + -0.75

Combine like terms: 2.561737691 + -0.75 = 1.811737691
t = 1.811737691

Simplifying
t = 1.811737691

Subproblem 2
t + 0.75 = -2.561737691

Simplifying
t + 0.75 = -2.561737691

Reorder the terms:
0.75 + t = -2.561737691

Solving
0.75 + t = -2.561737691

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 = -2.561737691 + -0.75

Combine like terms: 0.75 + -0.75 = 0.00
0.00 + t = -2.561737691 + -0.75
t = -2.561737691 + -0.75

Combine like terms: -2.561737691 + -0.75 = -3.311737691
t = -3.311737691

Simplifying
t = -3.311737691

Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {1.811737691, -3.311737691}
Solution
t = {1.811737691, -3.311737691}