3y^2-20y+15=0

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Solution for 3y^2-20y+15=0 equation: 


Simplifying
3y2 + -20y + 15 = 0

Reorder the terms:
15 + -20y + 3y2 = 0

Solving
15 + -20y + 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'.
5 + -6.666666667y + y2 = 0

Move the constant term to the right:

Add '-5' to each side of the equation.
5 + -6.666666667y + -5 + y2 = 0 + -5

Reorder the terms:
5 + -5 + -6.666666667y + y2 = 0 + -5

Combine like terms: 5 + -5 = 0
0 + -6.666666667y + y2 = 0 + -5
-6.666666667y + y2 = 0 + -5

Combine like terms: 0 + -5 = -5
-6.666666667y + y2 = -5

The y term is -6.666666667y.  Take half its coefficient (-3.333333334).
Square it (11.11111112) and add it to both sides.

Add '11.11111112' to each side of the equation.
-6.666666667y + 11.11111112 + y2 = -5 + 11.11111112

Reorder the terms:
11.11111112 + -6.666666667y + y2 = -5 + 11.11111112

Combine like terms: -5 + 11.11111112 = 6.11111112
11.11111112 + -6.666666667y + y2 = 6.11111112

Factor a perfect square on the left side:
(y + -3.333333334)(y + -3.333333334) = 6.11111112

Calculate the square root of the right side: 2.472066164

Break this problem into two subproblems by setting 
(y + -3.333333334) equal to 2.472066164 and -2.472066164.


Subproblem 1
y + -3.333333334 = 2.472066164

Simplifying
y + -3.333333334 = 2.472066164

Reorder the terms:
-3.333333334 + y = 2.472066164

Solving
-3.333333334 + y = 2.472066164

Solving for variable 'y'.

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

Add '3.333333334' to each side of the equation.
-3.333333334 + 3.333333334 + y = 2.472066164 + 3.333333334

Combine like terms: -3.333333334 + 3.333333334 = 0.000000000
0.000000000 + y = 2.472066164 + 3.333333334
y = 2.472066164 + 3.333333334

Combine like terms: 2.472066164 + 3.333333334 = 5.805399498
y = 5.805399498

Simplifying
y = 5.805399498


Subproblem 2
y + -3.333333334 = -2.472066164

Simplifying
y + -3.333333334 = -2.472066164

Reorder the terms:
-3.333333334 + y = -2.472066164

Solving
-3.333333334 + y = -2.472066164

Solving for variable 'y'.

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

Add '3.333333334' to each side of the equation.
-3.333333334 + 3.333333334 + y = -2.472066164 + 3.333333334

Combine like terms: -3.333333334 + 3.333333334 = 0.000000000
0.000000000 + y = -2.472066164 + 3.333333334
y = -2.472066164 + 3.333333334

Combine like terms: -2.472066164 + 3.333333334 = 0.86126717
y = 0.86126717

Simplifying
y = 0.86126717


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {5.805399498, 0.86126717}

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