9y^2+30y-32=0

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Solution for 9y^2+30y-32=0 equation: 


Simplifying
9y2 + 30y + -32 = 0

Reorder the terms:
-32 + 30y + 9y2 = 0

Solving
-32 + 30y + 9y2 = 0

Solving for variable 'y'.

Begin completing the square.  Divide all terms by
9 the coefficient of the squared term: 

Divide each side by '9'.
-3.555555556 + 3.333333333y + y2 = 0

Move the constant term to the right:

Add '3.555555556' to each side of the equation.
-3.555555556 + 3.333333333y + 3.555555556 + y2 = 0 + 3.555555556

Reorder the terms:
-3.555555556 + 3.555555556 + 3.333333333y + y2 = 0 + 3.555555556

Combine like terms: -3.555555556 + 3.555555556 = 0.000000000
0.000000000 + 3.333333333y + y2 = 0 + 3.555555556
3.333333333y + y2 = 0 + 3.555555556

Combine like terms: 0 + 3.555555556 = 3.555555556
3.333333333y + y2 = 3.555555556

The y term is 3.333333333y.  Take half its coefficient (1.666666667).
Square it (2.777777779) and add it to both sides.

Add '2.777777779' to each side of the equation.
3.333333333y + 2.777777779 + y2 = 3.555555556 + 2.777777779

Reorder the terms:
2.777777779 + 3.333333333y + y2 = 3.555555556 + 2.777777779

Combine like terms: 3.555555556 + 2.777777779 = 6.333333335
2.777777779 + 3.333333333y + y2 = 6.333333335

Factor a perfect square on the left side:
(y + 1.666666667)(y + 1.666666667) = 6.333333335

Calculate the square root of the right side: 2.516611479

Break this problem into two subproblems by setting 
(y + 1.666666667) equal to 2.516611479 and -2.516611479.


Subproblem 1
y + 1.666666667 = 2.516611479

Simplifying
y + 1.666666667 = 2.516611479

Reorder the terms:
1.666666667 + y = 2.516611479

Solving
1.666666667 + y = 2.516611479

Solving for variable 'y'.

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

Add '-1.666666667' to each side of the equation.
1.666666667 + -1.666666667 + y = 2.516611479 + -1.666666667

Combine like terms: 1.666666667 + -1.666666667 = 0.000000000
0.000000000 + y = 2.516611479 + -1.666666667
y = 2.516611479 + -1.666666667

Combine like terms: 2.516611479 + -1.666666667 = 0.849944812
y = 0.849944812

Simplifying
y = 0.849944812


Subproblem 2
y + 1.666666667 = -2.516611479

Simplifying
y + 1.666666667 = -2.516611479

Reorder the terms:
1.666666667 + y = -2.516611479

Solving
1.666666667 + y = -2.516611479

Solving for variable 'y'.

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

Add '-1.666666667' to each side of the equation.
1.666666667 + -1.666666667 + y = -2.516611479 + -1.666666667

Combine like terms: 1.666666667 + -1.666666667 = 0.000000000
0.000000000 + y = -2.516611479 + -1.666666667
y = -2.516611479 + -1.666666667

Combine like terms: -2.516611479 + -1.666666667 = -4.183278146
y = -4.183278146

Simplifying
y = -4.183278146


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {0.849944812, -4.183278146}

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