6y^2-31y-9=0

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Solution for 6y^2-31y-9=0 equation: 


Simplifying
6y2 + -31y + -9 = 0

Reorder the terms:
-9 + -31y + 6y2 = 0

Solving
-9 + -31y + 6y2 = 0

Solving for variable 'y'.

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

Divide each side by '6'.
-1.5 + -5.166666667y + y2 = 0

Move the constant term to the right:

Add '1.5' to each side of the equation.
-1.5 + -5.166666667y + 1.5 + y2 = 0 + 1.5

Reorder the terms:
-1.5 + 1.5 + -5.166666667y + y2 = 0 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + -5.166666667y + y2 = 0 + 1.5
-5.166666667y + y2 = 0 + 1.5

Combine like terms: 0 + 1.5 = 1.5
-5.166666667y + y2 = 1.5

The y term is -5.166666667y.  Take half its coefficient (-2.583333334).
Square it (6.673611115) and add it to both sides.

Add '6.673611115' to each side of the equation.
-5.166666667y + 6.673611115 + y2 = 1.5 + 6.673611115

Reorder the terms:
6.673611115 + -5.166666667y + y2 = 1.5 + 6.673611115

Combine like terms: 1.5 + 6.673611115 = 8.173611115
6.673611115 + -5.166666667y + y2 = 8.173611115

Factor a perfect square on the left side:
(y + -2.583333334)(y + -2.583333334) = 8.173611115

Calculate the square root of the right side: 2.8589528

Break this problem into two subproblems by setting 
(y + -2.583333334) equal to 2.8589528 and -2.8589528.


Subproblem 1
y + -2.583333334 = 2.8589528

Simplifying
y + -2.583333334 = 2.8589528

Reorder the terms:
-2.583333334 + y = 2.8589528

Solving
-2.583333334 + y = 2.8589528

Solving for variable 'y'.

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

Add '2.583333334' to each side of the equation.
-2.583333334 + 2.583333334 + y = 2.8589528 + 2.583333334

Combine like terms: -2.583333334 + 2.583333334 = 0.000000000
0.000000000 + y = 2.8589528 + 2.583333334
y = 2.8589528 + 2.583333334

Combine like terms: 2.8589528 + 2.583333334 = 5.442286134
y = 5.442286134

Simplifying
y = 5.442286134


Subproblem 2
y + -2.583333334 = -2.8589528

Simplifying
y + -2.583333334 = -2.8589528

Reorder the terms:
-2.583333334 + y = -2.8589528

Solving
-2.583333334 + y = -2.8589528

Solving for variable 'y'.

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

Add '2.583333334' to each side of the equation.
-2.583333334 + 2.583333334 + y = -2.8589528 + 2.583333334

Combine like terms: -2.583333334 + 2.583333334 = 0.000000000
0.000000000 + y = -2.8589528 + 2.583333334
y = -2.8589528 + 2.583333334

Combine like terms: -2.8589528 + 2.583333334 = -0.275619466
y = -0.275619466

Simplifying
y = -0.275619466


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {5.442286134, -0.275619466}

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