9y^2-30y+20=0

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Solution for 9y^2-30y+20=0 equation: 


Simplifying
9y2 + -30y + 20 = 0

Reorder the terms:
20 + -30y + 9y2 = 0

Solving
20 + -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'.
2.222222222 + -3.333333333y + y2 = 0

Move the constant term to the right:

Add '-2.222222222' to each side of the equation.
2.222222222 + -3.333333333y + -2.222222222 + y2 = 0 + -2.222222222

Reorder the terms:
2.222222222 + -2.222222222 + -3.333333333y + y2 = 0 + -2.222222222

Combine like terms: 2.222222222 + -2.222222222 = 0.000000000
0.000000000 + -3.333333333y + y2 = 0 + -2.222222222
-3.333333333y + y2 = 0 + -2.222222222

Combine like terms: 0 + -2.222222222 = -2.222222222
-3.333333333y + y2 = -2.222222222

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 = -2.222222222 + 2.777777779

Reorder the terms:
2.777777779 + -3.333333333y + y2 = -2.222222222 + 2.777777779

Combine like terms: -2.222222222 + 2.777777779 = 0.555555557
2.777777779 + -3.333333333y + y2 = 0.555555557

Factor a perfect square on the left side:
(y + -1.666666667)(y + -1.666666667) = 0.555555557

Calculate the square root of the right side: 0.745355993

Break this problem into two subproblems by setting 
(y + -1.666666667) equal to 0.745355993 and -0.745355993.


Subproblem 1
y + -1.666666667 = 0.745355993

Simplifying
y + -1.666666667 = 0.745355993

Reorder the terms:
-1.666666667 + y = 0.745355993

Solving
-1.666666667 + y = 0.745355993

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 = 0.745355993 + 1.666666667

Combine like terms: -1.666666667 + 1.666666667 = 0.000000000
0.000000000 + y = 0.745355993 + 1.666666667
y = 0.745355993 + 1.666666667

Combine like terms: 0.745355993 + 1.666666667 = 2.41202266
y = 2.41202266

Simplifying
y = 2.41202266


Subproblem 2
y + -1.666666667 = -0.745355993

Simplifying
y + -1.666666667 = -0.745355993

Reorder the terms:
-1.666666667 + y = -0.745355993

Solving
-1.666666667 + y = -0.745355993

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 = -0.745355993 + 1.666666667

Combine like terms: -1.666666667 + 1.666666667 = 0.000000000
0.000000000 + y = -0.745355993 + 1.666666667
y = -0.745355993 + 1.666666667

Combine like terms: -0.745355993 + 1.666666667 = 0.921310674
y = 0.921310674

Simplifying
y = 0.921310674


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {2.41202266, 0.921310674}

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