3y^2+y-32=0

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


Simplifying
3y2 + y + -32 = 0

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

Solving
-32 + y + 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 + 0.3333333333y + y2 = 0

Move the constant term to the right:

Add '10.66666667' to each side of the equation.
-10.66666667 + 0.3333333333y + 10.66666667 + y2 = 0 + 10.66666667

Reorder the terms:
-10.66666667 + 10.66666667 + 0.3333333333y + y2 = 0 + 10.66666667

Combine like terms: -10.66666667 + 10.66666667 = 0.00000000
0.00000000 + 0.3333333333y + y2 = 0 + 10.66666667
0.3333333333y + y2 = 0 + 10.66666667

Combine like terms: 0 + 10.66666667 = 10.66666667
0.3333333333y + y2 = 10.66666667

The y term is y.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
0.3333333333y + 0.25 + y2 = 10.66666667 + 0.25

Reorder the terms:
0.25 + 0.3333333333y + y2 = 10.66666667 + 0.25

Combine like terms: 10.66666667 + 0.25 = 10.91666667
0.25 + 0.3333333333y + y2 = 10.91666667

Factor a perfect square on the left side:
(y + 0.5)(y + 0.5) = 10.91666667

Calculate the square root of the right side: 3.304037934

Break this problem into two subproblems by setting 
(y + 0.5) equal to 3.304037934 and -3.304037934.


Subproblem 1
y + 0.5 = 3.304037934

Simplifying
y + 0.5 = 3.304037934

Reorder the terms:
0.5 + y = 3.304037934

Solving
0.5 + y = 3.304037934

Solving for variable 'y'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + y = 3.304037934 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + y = 3.304037934 + -0.5
y = 3.304037934 + -0.5

Combine like terms: 3.304037934 + -0.5 = 2.804037934
y = 2.804037934

Simplifying
y = 2.804037934


Subproblem 2
y + 0.5 = -3.304037934

Simplifying
y + 0.5 = -3.304037934

Reorder the terms:
0.5 + y = -3.304037934

Solving
0.5 + y = -3.304037934

Solving for variable 'y'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + y = -3.304037934 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + y = -3.304037934 + -0.5
y = -3.304037934 + -0.5

Combine like terms: -3.304037934 + -0.5 = -3.804037934
y = -3.804037934

Simplifying
y = -3.804037934


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {2.804037934, -3.804037934}

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