3y^2-11y-2=0

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Solution for 3y^2-11y-2=0 equation: 


Simplifying
3y2 + -11y + -2 = 0

Reorder the terms:
-2 + -11y + 3y2 = 0

Solving
-2 + -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'.
-0.6666666667 + -3.666666667y + y2 = 0

Move the constant term to the right:

Add '0.6666666667' to each side of the equation.
-0.6666666667 + -3.666666667y + 0.6666666667 + y2 = 0 + 0.6666666667

Reorder the terms:
-0.6666666667 + 0.6666666667 + -3.666666667y + y2 = 0 + 0.6666666667

Combine like terms: -0.6666666667 + 0.6666666667 = 0.0000000000
0.0000000000 + -3.666666667y + y2 = 0 + 0.6666666667
-3.666666667y + y2 = 0 + 0.6666666667

Combine like terms: 0 + 0.6666666667 = 0.6666666667
-3.666666667y + y2 = 0.6666666667

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 = 0.6666666667 + 3.361111114

Reorder the terms:
3.361111114 + -3.666666667y + y2 = 0.6666666667 + 3.361111114

Combine like terms: 0.6666666667 + 3.361111114 = 4.0277777807
3.361111114 + -3.666666667y + y2 = 4.0277777807

Factor a perfect square on the left side:
(y + -1.833333334)(y + -1.833333334) = 4.0277777807

Calculate the square root of the right side: 2.006932431

Break this problem into two subproblems by setting 
(y + -1.833333334) equal to 2.006932431 and -2.006932431.


Subproblem 1
y + -1.833333334 = 2.006932431

Simplifying
y + -1.833333334 = 2.006932431

Reorder the terms:
-1.833333334 + y = 2.006932431

Solving
-1.833333334 + y = 2.006932431

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 = 2.006932431 + 1.833333334

Combine like terms: -1.833333334 + 1.833333334 = 0.000000000
0.000000000 + y = 2.006932431 + 1.833333334
y = 2.006932431 + 1.833333334

Combine like terms: 2.006932431 + 1.833333334 = 3.840265765
y = 3.840265765

Simplifying
y = 3.840265765


Subproblem 2
y + -1.833333334 = -2.006932431

Simplifying
y + -1.833333334 = -2.006932431

Reorder the terms:
-1.833333334 + y = -2.006932431

Solving
-1.833333334 + y = -2.006932431

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 = -2.006932431 + 1.833333334

Combine like terms: -1.833333334 + 1.833333334 = 0.000000000
0.000000000 + y = -2.006932431 + 1.833333334
y = -2.006932431 + 1.833333334

Combine like terms: -2.006932431 + 1.833333334 = -0.173599097
y = -0.173599097

Simplifying
y = -0.173599097


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {3.840265765, -0.173599097}

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