3t(t+1)=9

Solution for 3t(t+1)=9 equation:

Simplifying
3t(t + 1) = 9

Reorder the terms:
3t(1 + t) = 9
(1 * 3t + t * 3t) = 9
(3t + 3t2) = 9

Solving
3t + 3t2 = 9

Solving for variable 't'.

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

Combine like terms: 9 + -9 = 0
-9 + 3t + 3t2 = 0

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

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

Simplifying
-3 + t + t2 = 0

Solving
-3 + t + t2 = 0

Begin completing the square.

Move the constant term to the right:

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

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

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

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

The t term is t.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
t + 0.25 + t2 = 3 + 0.25

Reorder the terms:
0.25 + t + t2 = 3 + 0.25

Combine like terms: 3 + 0.25 = 3.25
0.25 + t + t2 = 3.25

Factor a perfect square on the left side:
(t + 0.5)(t + 0.5) = 3.25

Calculate the square root of the right side: 1.802775638

Break this problem into two subproblems by setting 
(t + 0.5) equal to 1.802775638 and -1.802775638.

Subproblem 1
t + 0.5 = 1.802775638

Simplifying
t + 0.5 = 1.802775638

Reorder the terms:
0.5 + t = 1.802775638

Solving
0.5 + t = 1.802775638

Solving for variable 't'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + t = 1.802775638 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + t = 1.802775638 + -0.5
t = 1.802775638 + -0.5

Combine like terms: 1.802775638 + -0.5 = 1.302775638
t = 1.302775638

Simplifying
t = 1.302775638

Subproblem 2
t + 0.5 = -1.802775638

Simplifying
t + 0.5 = -1.802775638

Reorder the terms:
0.5 + t = -1.802775638

Solving
0.5 + t = -1.802775638

Solving for variable 't'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + t = -1.802775638 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + t = -1.802775638 + -0.5
t = -1.802775638 + -0.5

Combine like terms: -1.802775638 + -0.5 = -2.302775638
t = -2.302775638

Simplifying
t = -2.302775638

Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {1.302775638, -2.302775638}
Solution
t = {1.302775638, -2.302775638}