9y^2-30y+19=0

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Solution for 9y^2-30y+19=0 equation: 


Simplifying
9y2 + -30y + 19 = 0

Reorder the terms:
19 + -30y + 9y2 = 0

Solving
19 + -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'.
2.111111111 + -3.333333333y + y2 = 0

Move the constant term to the right:

Add '-2.111111111' to each side of the equation.
2.111111111 + -3.333333333y + -2.111111111 + y2 = 0 + -2.111111111

Reorder the terms:
2.111111111 + -2.111111111 + -3.333333333y + y2 = 0 + -2.111111111

Combine like terms: 2.111111111 + -2.111111111 = 0.000000000
0.000000000 + -3.333333333y + y2 = 0 + -2.111111111
-3.333333333y + y2 = 0 + -2.111111111

Combine like terms: 0 + -2.111111111 = -2.111111111
-3.333333333y + y2 = -2.111111111

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 = -2.111111111 + 2.777777779

Reorder the terms:
2.777777779 + -3.333333333y + y2 = -2.111111111 + 2.777777779

Combine like terms: -2.111111111 + 2.777777779 = 0.666666668
2.777777779 + -3.333333333y + y2 = 0.666666668

Factor a perfect square on the left side:
(y + -1.666666667)(y + -1.666666667) = 0.666666668

Calculate the square root of the right side: 0.816496582

Break this problem into two subproblems by setting 
(y + -1.666666667) equal to 0.816496582 and -0.816496582.


Subproblem 1
y + -1.666666667 = 0.816496582

Simplifying
y + -1.666666667 = 0.816496582

Reorder the terms:
-1.666666667 + y = 0.816496582

Solving
-1.666666667 + y = 0.816496582

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 = 0.816496582 + 1.666666667

Combine like terms: -1.666666667 + 1.666666667 = 0.000000000
0.000000000 + y = 0.816496582 + 1.666666667
y = 0.816496582 + 1.666666667

Combine like terms: 0.816496582 + 1.666666667 = 2.483163249
y = 2.483163249

Simplifying
y = 2.483163249


Subproblem 2
y + -1.666666667 = -0.816496582

Simplifying
y + -1.666666667 = -0.816496582

Reorder the terms:
-1.666666667 + y = -0.816496582

Solving
-1.666666667 + y = -0.816496582

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 = -0.816496582 + 1.666666667

Combine like terms: -1.666666667 + 1.666666667 = 0.000000000
0.000000000 + y = -0.816496582 + 1.666666667
y = -0.816496582 + 1.666666667

Combine like terms: -0.816496582 + 1.666666667 = 0.850170085
y = 0.850170085

Simplifying
y = 0.850170085


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {2.483163249, 0.850170085}

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