3y^2-40y+75=0

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Solution for 3y^2-40y+75=0 equation: 


Simplifying
3y2 + -40y + 75 = 0

Reorder the terms:
75 + -40y + 3y2 = 0

Solving
75 + -40y + 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'.
25 + -13.33333333y + y2 = 0

Move the constant term to the right:

Add '-25' to each side of the equation.
25 + -13.33333333y + -25 + y2 = 0 + -25

Reorder the terms:
25 + -25 + -13.33333333y + y2 = 0 + -25

Combine like terms: 25 + -25 = 0
0 + -13.33333333y + y2 = 0 + -25
-13.33333333y + y2 = 0 + -25

Combine like terms: 0 + -25 = -25
-13.33333333y + y2 = -25

The y term is -13.33333333y.  Take half its coefficient (-6.666666665).
Square it (44.44444442) and add it to both sides.

Add '44.44444442' to each side of the equation.
-13.33333333y + 44.44444442 + y2 = -25 + 44.44444442

Reorder the terms:
44.44444442 + -13.33333333y + y2 = -25 + 44.44444442

Combine like terms: -25 + 44.44444442 = 19.44444442
44.44444442 + -13.33333333y + y2 = 19.44444442

Factor a perfect square on the left side:
(y + -6.666666665)(y + -6.666666665) = 19.44444442

Calculate the square root of the right side: 4.409585516

Break this problem into two subproblems by setting 
(y + -6.666666665) equal to 4.409585516 and -4.409585516.


Subproblem 1
y + -6.666666665 = 4.409585516

Simplifying
y + -6.666666665 = 4.409585516

Reorder the terms:
-6.666666665 + y = 4.409585516

Solving
-6.666666665 + y = 4.409585516

Solving for variable 'y'.

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

Add '6.666666665' to each side of the equation.
-6.666666665 + 6.666666665 + y = 4.409585516 + 6.666666665

Combine like terms: -6.666666665 + 6.666666665 = 0.000000000
0.000000000 + y = 4.409585516 + 6.666666665
y = 4.409585516 + 6.666666665

Combine like terms: 4.409585516 + 6.666666665 = 11.076252181
y = 11.076252181

Simplifying
y = 11.076252181


Subproblem 2
y + -6.666666665 = -4.409585516

Simplifying
y + -6.666666665 = -4.409585516

Reorder the terms:
-6.666666665 + y = -4.409585516

Solving
-6.666666665 + y = -4.409585516

Solving for variable 'y'.

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

Add '6.666666665' to each side of the equation.
-6.666666665 + 6.666666665 + y = -4.409585516 + 6.666666665

Combine like terms: -6.666666665 + 6.666666665 = 0.000000000
0.000000000 + y = -4.409585516 + 6.666666665
y = -4.409585516 + 6.666666665

Combine like terms: -4.409585516 + 6.666666665 = 2.257081149
y = 2.257081149

Simplifying
y = 2.257081149


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {11.076252181, 2.257081149}

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