3y^2-8y-32=0

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Solution for 3y^2-8y-32=0 equation: 


Simplifying
3y2 + -8y + -32 = 0

Reorder the terms:
-32 + -8y + 3y2 = 0

Solving
-32 + -8y + 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'.
-10.66666667 + -2.666666667y + y2 = 0

Move the constant term to the right:

Add '10.66666667' to each side of the equation.
-10.66666667 + -2.666666667y + 10.66666667 + y2 = 0 + 10.66666667

Reorder the terms:
-10.66666667 + 10.66666667 + -2.666666667y + y2 = 0 + 10.66666667

Combine like terms: -10.66666667 + 10.66666667 = 0.00000000
0.00000000 + -2.666666667y + y2 = 0 + 10.66666667
-2.666666667y + y2 = 0 + 10.66666667

Combine like terms: 0 + 10.66666667 = 10.66666667
-2.666666667y + y2 = 10.66666667

The y term is -2.666666667y.  Take half its coefficient (-1.333333334).
Square it (1.777777780) and add it to both sides.

Add '1.777777780' to each side of the equation.
-2.666666667y + 1.777777780 + y2 = 10.66666667 + 1.777777780

Reorder the terms:
1.777777780 + -2.666666667y + y2 = 10.66666667 + 1.777777780

Combine like terms: 10.66666667 + 1.777777780 = 12.44444445
1.777777780 + -2.666666667y + y2 = 12.44444445

Factor a perfect square on the left side:
(y + -1.333333334)(y + -1.333333334) = 12.44444445

Calculate the square root of the right side: 3.527668416

Break this problem into two subproblems by setting 
(y + -1.333333334) equal to 3.527668416 and -3.527668416.


Subproblem 1
y + -1.333333334 = 3.527668416

Simplifying
y + -1.333333334 = 3.527668416

Reorder the terms:
-1.333333334 + y = 3.527668416

Solving
-1.333333334 + y = 3.527668416

Solving for variable 'y'.

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

Add '1.333333334' to each side of the equation.
-1.333333334 + 1.333333334 + y = 3.527668416 + 1.333333334

Combine like terms: -1.333333334 + 1.333333334 = 0.000000000
0.000000000 + y = 3.527668416 + 1.333333334
y = 3.527668416 + 1.333333334

Combine like terms: 3.527668416 + 1.333333334 = 4.86100175
y = 4.86100175

Simplifying
y = 4.86100175


Subproblem 2
y + -1.333333334 = -3.527668416

Simplifying
y + -1.333333334 = -3.527668416

Reorder the terms:
-1.333333334 + y = -3.527668416

Solving
-1.333333334 + y = -3.527668416

Solving for variable 'y'.

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

Add '1.333333334' to each side of the equation.
-1.333333334 + 1.333333334 + y = -3.527668416 + 1.333333334

Combine like terms: -1.333333334 + 1.333333334 = 0.000000000
0.000000000 + y = -3.527668416 + 1.333333334
y = -3.527668416 + 1.333333334

Combine like terms: -3.527668416 + 1.333333334 = -2.194335082
y = -2.194335082

Simplifying
y = -2.194335082


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {4.86100175, -2.194335082}

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