36y^2+24y+1=0

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Solution for 36y^2+24y+1=0 equation: 


Simplifying
36y2 + 24y + 1 = 0

Reorder the terms:
1 + 24y + 36y2 = 0

Solving
1 + 24y + 36y2 = 0

Solving for variable 'y'.

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

Divide each side by '36'.
0.02777777778 + 0.6666666667y + y2 = 0

Move the constant term to the right:

Add '-0.02777777778' to each side of the equation.
0.02777777778 + 0.6666666667y + -0.02777777778 + y2 = 0 + -0.02777777778

Reorder the terms:
0.02777777778 + -0.02777777778 + 0.6666666667y + y2 = 0 + -0.02777777778

Combine like terms: 0.02777777778 + -0.02777777778 = 0.00000000000
0.00000000000 + 0.6666666667y + y2 = 0 + -0.02777777778
0.6666666667y + y2 = 0 + -0.02777777778

Combine like terms: 0 + -0.02777777778 = -0.02777777778
0.6666666667y + y2 = -0.02777777778

The y term is 0.6666666667y.  Take half its coefficient (0.3333333334).
Square it (0.1111111112) and add it to both sides.

Add '0.1111111112' to each side of the equation.
0.6666666667y + 0.1111111112 + y2 = -0.02777777778 + 0.1111111112

Reorder the terms:
0.1111111112 + 0.6666666667y + y2 = -0.02777777778 + 0.1111111112

Combine like terms: -0.02777777778 + 0.1111111112 = 0.08333333342
0.1111111112 + 0.6666666667y + y2 = 0.08333333342

Factor a perfect square on the left side:
(y + 0.3333333334)(y + 0.3333333334) = 0.08333333342

Calculate the square root of the right side: 0.288675135

Break this problem into two subproblems by setting 
(y + 0.3333333334) equal to 0.288675135 and -0.288675135.


Subproblem 1
y + 0.3333333334 = 0.288675135

Simplifying
y + 0.3333333334 = 0.288675135

Reorder the terms:
0.3333333334 + y = 0.288675135

Solving
0.3333333334 + y = 0.288675135

Solving for variable 'y'.

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

Add '-0.3333333334' to each side of the equation.
0.3333333334 + -0.3333333334 + y = 0.288675135 + -0.3333333334

Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000
0.0000000000 + y = 0.288675135 + -0.3333333334
y = 0.288675135 + -0.3333333334

Combine like terms: 0.288675135 + -0.3333333334 = -0.0446581984
y = -0.0446581984

Simplifying
y = -0.0446581984


Subproblem 2
y + 0.3333333334 = -0.288675135

Simplifying
y + 0.3333333334 = -0.288675135

Reorder the terms:
0.3333333334 + y = -0.288675135

Solving
0.3333333334 + y = -0.288675135

Solving for variable 'y'.

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

Add '-0.3333333334' to each side of the equation.
0.3333333334 + -0.3333333334 + y = -0.288675135 + -0.3333333334

Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000
0.0000000000 + y = -0.288675135 + -0.3333333334
y = -0.288675135 + -0.3333333334

Combine like terms: -0.288675135 + -0.3333333334 = -0.6220084684
y = -0.6220084684

Simplifying
y = -0.6220084684


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {-0.0446581984, -0.6220084684}

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