3y^2+11y+9=0

Solution for 3y^2+11y+9=0 equation:

Simplifying
3y2 + 11y + 9 = 0

Reorder the terms:
9 + 11y + 3y2 = 0

Solving
9 + 11y + 3y2 = 0

Solving for variable 'y'.

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

Divide each side by '3'.
3 + 3.666666667y + y2 = 0

Move the constant term to the right:

Add '-3' to each side of the equation.
3 + 3.666666667y + -3 + y2 = 0 + -3

Reorder the terms:
3 + -3 + 3.666666667y + y2 = 0 + -3

Combine like terms: 3 + -3 = 0
0 + 3.666666667y + y2 = 0 + -3
3.666666667y + y2 = 0 + -3

Combine like terms: 0 + -3 = -3
3.666666667y + y2 = -3

The y term is 3.666666667y.  Take half its coefficient (1.833333334).
Square it (3.361111114) and add it to both sides.

Add '3.361111114' to each side of the equation.
3.666666667y + 3.361111114 + y2 = -3 + 3.361111114

Reorder the terms:
3.361111114 + 3.666666667y + y2 = -3 + 3.361111114

Combine like terms: -3 + 3.361111114 = 0.361111114
3.361111114 + 3.666666667y + y2 = 0.361111114

Factor a perfect square on the left side:
(y + 1.833333334)(y + 1.833333334) = 0.361111114

Calculate the square root of the right side: 0.600925215

Break this problem into two subproblems by setting 
(y + 1.833333334) equal to 0.600925215 and -0.600925215.

Subproblem 1
y + 1.833333334 = 0.600925215

Simplifying
y + 1.833333334 = 0.600925215

Reorder the terms:
1.833333334 + y = 0.600925215

Solving
1.833333334 + y = 0.600925215

Solving for variable 'y'.

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

Add '-1.833333334' to each side of the equation.
1.833333334 + -1.833333334 + y = 0.600925215 + -1.833333334

Combine like terms: 1.833333334 + -1.833333334 = 0.000000000
0.000000000 + y = 0.600925215 + -1.833333334
y = 0.600925215 + -1.833333334

Combine like terms: 0.600925215 + -1.833333334 = -1.232408119
y = -1.232408119

Simplifying
y = -1.232408119

Subproblem 2
y + 1.833333334 = -0.600925215

Simplifying
y + 1.833333334 = -0.600925215

Reorder the terms:
1.833333334 + y = -0.600925215

Solving
1.833333334 + y = -0.600925215

Solving for variable 'y'.

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

Add '-1.833333334' to each side of the equation.
1.833333334 + -1.833333334 + y = -0.600925215 + -1.833333334

Combine like terms: 1.833333334 + -1.833333334 = 0.000000000
0.000000000 + y = -0.600925215 + -1.833333334
y = -0.600925215 + -1.833333334

Combine like terms: -0.600925215 + -1.833333334 = -2.434258549
y = -2.434258549

Simplifying
y = -2.434258549

Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {-1.232408119, -2.434258549}