3t+3t(t+8.2)=37.5

Solution for 3t+3t(t+8.2)=37.5 equation:

Simplifying
3t + 3t(t + 8.2) = 37.5

Reorder the terms:
3t + 3t(8.2 + t) = 37.5
3t + (8.2 * 3t + t * 3t) = 37.5
3t + (24.6t + 3t2) = 37.5

Combine like terms: 3t + 24.6t = 27.6t
27.6t + 3t2 = 37.5

Solving
27.6t + 3t2 = 37.5

Solving for variable 't'.

Reorder the terms:
-37.5 + 27.6t + 3t2 = 37.5 + -37.5

Combine like terms: 37.5 + -37.5 = 0.0
-37.5 + 27.6t + 3t2 = 0.0

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

Divide each side by '3'.
-12.5 + 9.2t + t2 = 0.0

Move the constant term to the right:

Add '12.5' to each side of the equation.
-12.5 + 9.2t + 12.5 + t2 = 0.0 + 12.5

Reorder the terms:
-12.5 + 12.5 + 9.2t + t2 = 0.0 + 12.5

Combine like terms: -12.5 + 12.5 = 0.0
0.0 + 9.2t + t2 = 0.0 + 12.5
9.2t + t2 = 0.0 + 12.5

Combine like terms: 0.0 + 12.5 = 12.5
9.2t + t2 = 12.5

The t term is 9.2t.  Take half its coefficient (4.6).
Square it (21.16) and add it to both sides.

Add '21.16' to each side of the equation.
9.2t + 21.16 + t2 = 12.5 + 21.16

Reorder the terms:
21.16 + 9.2t + t2 = 12.5 + 21.16

Combine like terms: 12.5 + 21.16 = 33.66
21.16 + 9.2t + t2 = 33.66

Factor a perfect square on the left side:
(t + 4.6)(t + 4.6) = 33.66

Calculate the square root of the right side: 5.801723882

Break this problem into two subproblems by setting 
(t + 4.6) equal to 5.801723882 and -5.801723882.

Subproblem 1
t + 4.6 = 5.801723882

Simplifying
t + 4.6 = 5.801723882

Reorder the terms:
4.6 + t = 5.801723882

Solving
4.6 + t = 5.801723882

Solving for variable 't'.

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

Add '-4.6' to each side of the equation.
4.6 + -4.6 + t = 5.801723882 + -4.6

Combine like terms: 4.6 + -4.6 = 0.0
0.0 + t = 5.801723882 + -4.6
t = 5.801723882 + -4.6

Combine like terms: 5.801723882 + -4.6 = 1.201723882
t = 1.201723882

Simplifying
t = 1.201723882

Subproblem 2
t + 4.6 = -5.801723882

Simplifying
t + 4.6 = -5.801723882

Reorder the terms:
4.6 + t = -5.801723882

Solving
4.6 + t = -5.801723882

Solving for variable 't'.

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

Add '-4.6' to each side of the equation.
4.6 + -4.6 + t = -5.801723882 + -4.6

Combine like terms: 4.6 + -4.6 = 0.0
0.0 + t = -5.801723882 + -4.6
t = -5.801723882 + -4.6

Combine like terms: -5.801723882 + -4.6 = -10.401723882
t = -10.401723882

Simplifying
t = -10.401723882

Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {1.201723882, -10.401723882}