3y^2+2y-6=0

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Solution for 3y^2+2y-6=0 equation: 


Simplifying
3y2 + 2y + -6 = 0

Reorder the terms:
-6 + 2y + 3y2 = 0

Solving
-6 + 2y + 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'.
-2 + 0.6666666667y + y2 = 0

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + 0.6666666667y + 2 + y2 = 0 + 2

Reorder the terms:
-2 + 2 + 0.6666666667y + y2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + 0.6666666667y + y2 = 0 + 2
0.6666666667y + y2 = 0 + 2

Combine like terms: 0 + 2 = 2
0.6666666667y + y2 = 2

The y term is 0.6666666667y.  Take half its coefficient (0.3333333334).
Square it (0.1111111112) and add it to both sides.

Add '0.1111111112' to each side of the equation.
0.6666666667y + 0.1111111112 + y2 = 2 + 0.1111111112

Reorder the terms:
0.1111111112 + 0.6666666667y + y2 = 2 + 0.1111111112

Combine like terms: 2 + 0.1111111112 = 2.1111111112
0.1111111112 + 0.6666666667y + y2 = 2.1111111112

Factor a perfect square on the left side:
(y + 0.3333333334)(y + 0.3333333334) = 2.1111111112

Calculate the square root of the right side: 1.452966315

Break this problem into two subproblems by setting 
(y + 0.3333333334) equal to 1.452966315 and -1.452966315.


Subproblem 1
y + 0.3333333334 = 1.452966315

Simplifying
y + 0.3333333334 = 1.452966315

Reorder the terms:
0.3333333334 + y = 1.452966315

Solving
0.3333333334 + y = 1.452966315

Solving for variable 'y'.

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

Add '-0.3333333334' to each side of the equation.
0.3333333334 + -0.3333333334 + y = 1.452966315 + -0.3333333334

Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000
0.0000000000 + y = 1.452966315 + -0.3333333334
y = 1.452966315 + -0.3333333334

Combine like terms: 1.452966315 + -0.3333333334 = 1.1196329816
y = 1.1196329816

Simplifying
y = 1.1196329816


Subproblem 2
y + 0.3333333334 = -1.452966315

Simplifying
y + 0.3333333334 = -1.452966315

Reorder the terms:
0.3333333334 + y = -1.452966315

Solving
0.3333333334 + y = -1.452966315

Solving for variable 'y'.

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

Add '-0.3333333334' to each side of the equation.
0.3333333334 + -0.3333333334 + y = -1.452966315 + -0.3333333334

Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000
0.0000000000 + y = -1.452966315 + -0.3333333334
y = -1.452966315 + -0.3333333334

Combine like terms: -1.452966315 + -0.3333333334 = -1.7862996484
y = -1.7862996484

Simplifying
y = -1.7862996484


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {1.1196329816, -1.7862996484}

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