10y+3y^2=30

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Solution for 10y+3y^2=30 equation: 


Simplifying
10y + 3y2 = 30

Solving
10y + 3y2 = 30

Solving for variable 'y'.

Reorder the terms:
-30 + 10y + 3y2 = 30 + -30

Combine like terms: 30 + -30 = 0
-30 + 10y + 3y2 = 0

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

Divide each side by '3'.
-10 + 3.333333333y + y2 = 0

Move the constant term to the right:

Add '10' to each side of the equation.
-10 + 3.333333333y + 10 + y2 = 0 + 10

Reorder the terms:
-10 + 10 + 3.333333333y + y2 = 0 + 10

Combine like terms: -10 + 10 = 0
0 + 3.333333333y + y2 = 0 + 10
3.333333333y + y2 = 0 + 10

Combine like terms: 0 + 10 = 10
3.333333333y + y2 = 10

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 = 10 + 2.777777779

Reorder the terms:
2.777777779 + 3.333333333y + y2 = 10 + 2.777777779

Combine like terms: 10 + 2.777777779 = 12.777777779
2.777777779 + 3.333333333y + y2 = 12.777777779

Factor a perfect square on the left side:
(y + 1.666666667)(y + 1.666666667) = 12.777777779

Calculate the square root of the right side: 3.574601765

Break this problem into two subproblems by setting 
(y + 1.666666667) equal to 3.574601765 and -3.574601765.


Subproblem 1
y + 1.666666667 = 3.574601765

Simplifying
y + 1.666666667 = 3.574601765

Reorder the terms:
1.666666667 + y = 3.574601765

Solving
1.666666667 + y = 3.574601765

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 = 3.574601765 + -1.666666667

Combine like terms: 1.666666667 + -1.666666667 = 0.000000000
0.000000000 + y = 3.574601765 + -1.666666667
y = 3.574601765 + -1.666666667

Combine like terms: 3.574601765 + -1.666666667 = 1.907935098
y = 1.907935098

Simplifying
y = 1.907935098


Subproblem 2
y + 1.666666667 = -3.574601765

Simplifying
y + 1.666666667 = -3.574601765

Reorder the terms:
1.666666667 + y = -3.574601765

Solving
1.666666667 + y = -3.574601765

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 = -3.574601765 + -1.666666667

Combine like terms: 1.666666667 + -1.666666667 = 0.000000000
0.000000000 + y = -3.574601765 + -1.666666667
y = -3.574601765 + -1.666666667

Combine like terms: -3.574601765 + -1.666666667 = -5.241268432
y = -5.241268432

Simplifying
y = -5.241268432


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {1.907935098, -5.241268432}

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