6y^2+13y-3=0

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Solution for 6y^2+13y-3=0 equation: 


Simplifying
6y2 + 13y + -3 = 0

Reorder the terms:
-3 + 13y + 6y2 = 0

Solving
-3 + 13y + 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'.
-0.5 + 2.166666667y + y2 = 0

Move the constant term to the right:

Add '0.5' to each side of the equation.
-0.5 + 2.166666667y + 0.5 + y2 = 0 + 0.5

Reorder the terms:
-0.5 + 0.5 + 2.166666667y + y2 = 0 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + 2.166666667y + y2 = 0 + 0.5
2.166666667y + y2 = 0 + 0.5

Combine like terms: 0 + 0.5 = 0.5
2.166666667y + y2 = 0.5

The y term is 2.166666667y.  Take half its coefficient (1.083333334).
Square it (1.173611113) and add it to both sides.

Add '1.173611113' to each side of the equation.
2.166666667y + 1.173611113 + y2 = 0.5 + 1.173611113

Reorder the terms:
1.173611113 + 2.166666667y + y2 = 0.5 + 1.173611113

Combine like terms: 0.5 + 1.173611113 = 1.673611113
1.173611113 + 2.166666667y + y2 = 1.673611113

Factor a perfect square on the left side:
(y + 1.083333334)(y + 1.083333334) = 1.673611113

Calculate the square root of the right side: 1.293681225

Break this problem into two subproblems by setting 
(y + 1.083333334) equal to 1.293681225 and -1.293681225.


Subproblem 1
y + 1.083333334 = 1.293681225

Simplifying
y + 1.083333334 = 1.293681225

Reorder the terms:
1.083333334 + y = 1.293681225

Solving
1.083333334 + y = 1.293681225

Solving for variable 'y'.

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

Add '-1.083333334' to each side of the equation.
1.083333334 + -1.083333334 + y = 1.293681225 + -1.083333334

Combine like terms: 1.083333334 + -1.083333334 = 0.000000000
0.000000000 + y = 1.293681225 + -1.083333334
y = 1.293681225 + -1.083333334

Combine like terms: 1.293681225 + -1.083333334 = 0.210347891
y = 0.210347891

Simplifying
y = 0.210347891


Subproblem 2
y + 1.083333334 = -1.293681225

Simplifying
y + 1.083333334 = -1.293681225

Reorder the terms:
1.083333334 + y = -1.293681225

Solving
1.083333334 + y = -1.293681225

Solving for variable 'y'.

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

Add '-1.083333334' to each side of the equation.
1.083333334 + -1.083333334 + y = -1.293681225 + -1.083333334

Combine like terms: 1.083333334 + -1.083333334 = 0.000000000
0.000000000 + y = -1.293681225 + -1.083333334
y = -1.293681225 + -1.083333334

Combine like terms: -1.293681225 + -1.083333334 = -2.377014559
y = -2.377014559

Simplifying
y = -2.377014559


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {0.210347891, -2.377014559}

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