3y^2+4y-9=0

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Solution for 3y^2+4y-9=0 equation: 


Simplifying
3y2 + 4y + -9 = 0

Reorder the terms:
-9 + 4y + 3y2 = 0

Solving
-9 + 4y + 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 + 1.333333333y + y2 = 0

Move the constant term to the right:

Add '3' to each side of the equation.
-3 + 1.333333333y + 3 + y2 = 0 + 3

Reorder the terms:
-3 + 3 + 1.333333333y + y2 = 0 + 3

Combine like terms: -3 + 3 = 0
0 + 1.333333333y + y2 = 0 + 3
1.333333333y + y2 = 0 + 3

Combine like terms: 0 + 3 = 3
1.333333333y + y2 = 3

The y term is 1.333333333y.  Take half its coefficient (0.6666666665).
Square it (0.4444444442) and add it to both sides.

Add '0.4444444442' to each side of the equation.
1.333333333y + 0.4444444442 + y2 = 3 + 0.4444444442

Reorder the terms:
0.4444444442 + 1.333333333y + y2 = 3 + 0.4444444442

Combine like terms: 3 + 0.4444444442 = 3.4444444442
0.4444444442 + 1.333333333y + y2 = 3.4444444442

Factor a perfect square on the left side:
(y + 0.6666666665)(y + 0.6666666665) = 3.4444444442

Calculate the square root of the right side: 1.855921454

Break this problem into two subproblems by setting 
(y + 0.6666666665) equal to 1.855921454 and -1.855921454.


Subproblem 1
y + 0.6666666665 = 1.855921454

Simplifying
y + 0.6666666665 = 1.855921454

Reorder the terms:
0.6666666665 + y = 1.855921454

Solving
0.6666666665 + y = 1.855921454

Solving for variable 'y'.

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

Add '-0.6666666665' to each side of the equation.
0.6666666665 + -0.6666666665 + y = 1.855921454 + -0.6666666665

Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000
0.0000000000 + y = 1.855921454 + -0.6666666665
y = 1.855921454 + -0.6666666665

Combine like terms: 1.855921454 + -0.6666666665 = 1.1892547875
y = 1.1892547875

Simplifying
y = 1.1892547875


Subproblem 2
y + 0.6666666665 = -1.855921454

Simplifying
y + 0.6666666665 = -1.855921454

Reorder the terms:
0.6666666665 + y = -1.855921454

Solving
0.6666666665 + y = -1.855921454

Solving for variable 'y'.

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

Add '-0.6666666665' to each side of the equation.
0.6666666665 + -0.6666666665 + y = -1.855921454 + -0.6666666665

Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000
0.0000000000 + y = -1.855921454 + -0.6666666665
y = -1.855921454 + -0.6666666665

Combine like terms: -1.855921454 + -0.6666666665 = -2.5225881205
y = -2.5225881205

Simplifying
y = -2.5225881205


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {1.1892547875, -2.5225881205}

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