9y^2-30y-216=0

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Solution for 9y^2-30y-216=0 equation: 


Simplifying
9y2 + -30y + -216 = 0

Reorder the terms:
-216 + -30y + 9y2 = 0

Solving
-216 + -30y + 9y2 = 0

Solving for variable 'y'.

Factor out the Greatest Common Factor (GCF), '3'.
3(-72 + -10y + 3y2) = 0

Ignore the factor 3.

Subproblem 1
Set the factor '(-72 + -10y + 3y2)' equal to zero and attempt to solve:

Simplifying
-72 + -10y + 3y2 = 0

Solving
-72 + -10y + 3y2 = 0

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

Divide each side by '3'.
-24 + -3.333333333y + y2 = 0

Move the constant term to the right:

Add '24' to each side of the equation.
-24 + -3.333333333y + 24 + y2 = 0 + 24

Reorder the terms:
-24 + 24 + -3.333333333y + y2 = 0 + 24

Combine like terms: -24 + 24 = 0
0 + -3.333333333y + y2 = 0 + 24
-3.333333333y + y2 = 0 + 24

Combine like terms: 0 + 24 = 24
-3.333333333y + y2 = 24

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 = 24 + 2.777777779

Reorder the terms:
2.777777779 + -3.333333333y + y2 = 24 + 2.777777779

Combine like terms: 24 + 2.777777779 = 26.777777779
2.777777779 + -3.333333333y + y2 = 26.777777779

Factor a perfect square on the left side:
(y + -1.666666667)(y + -1.666666667) = 26.777777779

Calculate the square root of the right side: 5.174724899

Break this problem into two subproblems by setting 
(y + -1.666666667) equal to 5.174724899 and -5.174724899.


Subproblem 1
y + -1.666666667 = 5.174724899

Simplifying
y + -1.666666667 = 5.174724899

Reorder the terms:
-1.666666667 + y = 5.174724899

Solving
-1.666666667 + y = 5.174724899

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 = 5.174724899 + 1.666666667

Combine like terms: -1.666666667 + 1.666666667 = 0.000000000
0.000000000 + y = 5.174724899 + 1.666666667
y = 5.174724899 + 1.666666667

Combine like terms: 5.174724899 + 1.666666667 = 6.841391566
y = 6.841391566

Simplifying
y = 6.841391566


Subproblem 2
y + -1.666666667 = -5.174724899

Simplifying
y + -1.666666667 = -5.174724899

Reorder the terms:
-1.666666667 + y = -5.174724899

Solving
-1.666666667 + y = -5.174724899

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 = -5.174724899 + 1.666666667

Combine like terms: -1.666666667 + 1.666666667 = 0.000000000
0.000000000 + y = -5.174724899 + 1.666666667
y = -5.174724899 + 1.666666667

Combine like terms: -5.174724899 + 1.666666667 = -3.508058232
y = -3.508058232

Simplifying
y = -3.508058232


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {6.841391566, -3.508058232}
Solution
y = {6.841391566, -3.508058232}

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