3y^2-23y-12=0

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Solution for 3y^2-23y-12=0 equation: 


Simplifying
3y2 + -23y + -12 = 0

Reorder the terms:
-12 + -23y + 3y2 = 0

Solving
-12 + -23y + 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'.
-4 + -7.666666667y + y2 = 0

Move the constant term to the right:

Add '4' to each side of the equation.
-4 + -7.666666667y + 4 + y2 = 0 + 4

Reorder the terms:
-4 + 4 + -7.666666667y + y2 = 0 + 4

Combine like terms: -4 + 4 = 0
0 + -7.666666667y + y2 = 0 + 4
-7.666666667y + y2 = 0 + 4

Combine like terms: 0 + 4 = 4
-7.666666667y + y2 = 4

The y term is -7.666666667y.  Take half its coefficient (-3.833333334).
Square it (14.69444445) and add it to both sides.

Add '14.69444445' to each side of the equation.
-7.666666667y + 14.69444445 + y2 = 4 + 14.69444445

Reorder the terms:
14.69444445 + -7.666666667y + y2 = 4 + 14.69444445

Combine like terms: 4 + 14.69444445 = 18.69444445
14.69444445 + -7.666666667y + y2 = 18.69444445

Factor a perfect square on the left side:
(y + -3.833333334)(y + -3.833333334) = 18.69444445

Calculate the square root of the right side: 4.323707258

Break this problem into two subproblems by setting 
(y + -3.833333334) equal to 4.323707258 and -4.323707258.


Subproblem 1
y + -3.833333334 = 4.323707258

Simplifying
y + -3.833333334 = 4.323707258

Reorder the terms:
-3.833333334 + y = 4.323707258

Solving
-3.833333334 + y = 4.323707258

Solving for variable 'y'.

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

Add '3.833333334' to each side of the equation.
-3.833333334 + 3.833333334 + y = 4.323707258 + 3.833333334

Combine like terms: -3.833333334 + 3.833333334 = 0.000000000
0.000000000 + y = 4.323707258 + 3.833333334
y = 4.323707258 + 3.833333334

Combine like terms: 4.323707258 + 3.833333334 = 8.157040592
y = 8.157040592

Simplifying
y = 8.157040592


Subproblem 2
y + -3.833333334 = -4.323707258

Simplifying
y + -3.833333334 = -4.323707258

Reorder the terms:
-3.833333334 + y = -4.323707258

Solving
-3.833333334 + y = -4.323707258

Solving for variable 'y'.

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

Add '3.833333334' to each side of the equation.
-3.833333334 + 3.833333334 + y = -4.323707258 + 3.833333334

Combine like terms: -3.833333334 + 3.833333334 = 0.000000000
0.000000000 + y = -4.323707258 + 3.833333334
y = -4.323707258 + 3.833333334

Combine like terms: -4.323707258 + 3.833333334 = -0.490373924
y = -0.490373924

Simplifying
y = -0.490373924


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {8.157040592, -0.490373924}

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