3y^2-2y-24=0

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Solution for 3y^2-2y-24=0 equation: 


Simplifying
3y2 + -2y + -24 = 0

Reorder the terms:
-24 + -2y + 3y2 = 0

Solving
-24 + -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'.
-8 + -0.6666666667y + y2 = 0

Move the constant term to the right:

Add '8' to each side of the equation.
-8 + -0.6666666667y + 8 + y2 = 0 + 8

Reorder the terms:
-8 + 8 + -0.6666666667y + y2 = 0 + 8

Combine like terms: -8 + 8 = 0
0 + -0.6666666667y + y2 = 0 + 8
-0.6666666667y + y2 = 0 + 8

Combine like terms: 0 + 8 = 8
-0.6666666667y + y2 = 8

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 = 8 + 0.1111111112

Reorder the terms:
0.1111111112 + -0.6666666667y + y2 = 8 + 0.1111111112

Combine like terms: 8 + 0.1111111112 = 8.1111111112
0.1111111112 + -0.6666666667y + y2 = 8.1111111112

Factor a perfect square on the left side:
(y + -0.3333333334)(y + -0.3333333334) = 8.1111111112

Calculate the square root of the right side: 2.848001248

Break this problem into two subproblems by setting 
(y + -0.3333333334) equal to 2.848001248 and -2.848001248.


Subproblem 1
y + -0.3333333334 = 2.848001248

Simplifying
y + -0.3333333334 = 2.848001248

Reorder the terms:
-0.3333333334 + y = 2.848001248

Solving
-0.3333333334 + y = 2.848001248

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 = 2.848001248 + 0.3333333334

Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000
0.0000000000 + y = 2.848001248 + 0.3333333334
y = 2.848001248 + 0.3333333334

Combine like terms: 2.848001248 + 0.3333333334 = 3.1813345814
y = 3.1813345814

Simplifying
y = 3.1813345814


Subproblem 2
y + -0.3333333334 = -2.848001248

Simplifying
y + -0.3333333334 = -2.848001248

Reorder the terms:
-0.3333333334 + y = -2.848001248

Solving
-0.3333333334 + y = -2.848001248

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 = -2.848001248 + 0.3333333334

Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000
0.0000000000 + y = -2.848001248 + 0.3333333334
y = -2.848001248 + 0.3333333334

Combine like terms: -2.848001248 + 0.3333333334 = -2.5146679146
y = -2.5146679146

Simplifying
y = -2.5146679146


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {3.1813345814, -2.5146679146}

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