2t^2+9t+6=0

Solution for 2t^2+9t+6=0 equation:

Simplifying
2t2 + 9t + 6 = 0

Reorder the terms:
6 + 9t + 2t2 = 0

Solving
6 + 9t + 2t2 = 0

Solving for variable 't'.

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

Divide each side by '2'.
3 + 4.5t + t2 = 0

Move the constant term to the right:

Add '-3' to each side of the equation.
3 + 4.5t + -3 + t2 = 0 + -3

Reorder the terms:
3 + -3 + 4.5t + t2 = 0 + -3

Combine like terms: 3 + -3 = 0
0 + 4.5t + t2 = 0 + -3
4.5t + t2 = 0 + -3

Combine like terms: 0 + -3 = -3
4.5t + t2 = -3

The t term is 4.5t.  Take half its coefficient (2.25).
Square it (5.0625) and add it to both sides.

Add '5.0625' to each side of the equation.
4.5t + 5.0625 + t2 = -3 + 5.0625

Reorder the terms:
5.0625 + 4.5t + t2 = -3 + 5.0625

Combine like terms: -3 + 5.0625 = 2.0625
5.0625 + 4.5t + t2 = 2.0625

Factor a perfect square on the left side:
(t + 2.25)(t + 2.25) = 2.0625

Calculate the square root of the right side: 1.436140662

Break this problem into two subproblems by setting 
(t + 2.25) equal to 1.436140662 and -1.436140662.

Subproblem 1
t + 2.25 = 1.436140662

Simplifying
t + 2.25 = 1.436140662

Reorder the terms:
2.25 + t = 1.436140662

Solving
2.25 + t = 1.436140662

Solving for variable 't'.

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

Add '-2.25' to each side of the equation.
2.25 + -2.25 + t = 1.436140662 + -2.25

Combine like terms: 2.25 + -2.25 = 0.00
0.00 + t = 1.436140662 + -2.25
t = 1.436140662 + -2.25

Combine like terms: 1.436140662 + -2.25 = -0.813859338
t = -0.813859338

Simplifying
t = -0.813859338

Subproblem 2
t + 2.25 = -1.436140662

Simplifying
t + 2.25 = -1.436140662

Reorder the terms:
2.25 + t = -1.436140662

Solving
2.25 + t = -1.436140662

Solving for variable 't'.

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

Add '-2.25' to each side of the equation.
2.25 + -2.25 + t = -1.436140662 + -2.25

Combine like terms: 2.25 + -2.25 = 0.00
0.00 + t = -1.436140662 + -2.25
t = -1.436140662 + -2.25

Combine like terms: -1.436140662 + -2.25 = -3.686140662
t = -3.686140662

Simplifying
t = -3.686140662

Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {-0.813859338, -3.686140662}