192y^2+306y+54=0

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Solution for 192y^2+306y+54=0 equation: 


Simplifying
192y2 + 306y + 54 = 0

Reorder the terms:
54 + 306y + 192y2 = 0

Solving
54 + 306y + 192y2 = 0

Solving for variable 'y'.

Factor out the Greatest Common Factor (GCF), '6'.
6(9 + 51y + 32y2) = 0

Ignore the factor 6.

Subproblem 1
Set the factor '(9 + 51y + 32y2)' equal to zero and attempt to solve:

Simplifying
9 + 51y + 32y2 = 0

Solving
9 + 51y + 32y2 = 0

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

Divide each side by '32'.
0.28125 + 1.59375y + y2 = 0

Move the constant term to the right:

Add '-0.28125' to each side of the equation.
0.28125 + 1.59375y + -0.28125 + y2 = 0 + -0.28125

Reorder the terms:
0.28125 + -0.28125 + 1.59375y + y2 = 0 + -0.28125

Combine like terms: 0.28125 + -0.28125 = 0.00000
0.00000 + 1.59375y + y2 = 0 + -0.28125
1.59375y + y2 = 0 + -0.28125

Combine like terms: 0 + -0.28125 = -0.28125
1.59375y + y2 = -0.28125

The y term is 1.59375y.  Take half its coefficient (0.796875).
Square it (0.6350097656) and add it to both sides.

Add '0.6350097656' to each side of the equation.
1.59375y + 0.6350097656 + y2 = -0.28125 + 0.6350097656

Reorder the terms:
0.6350097656 + 1.59375y + y2 = -0.28125 + 0.6350097656

Combine like terms: -0.28125 + 0.6350097656 = 0.3537597656
0.6350097656 + 1.59375y + y2 = 0.3537597656

Factor a perfect square on the left side:
(y + 0.796875)(y + 0.796875) = 0.3537597656

Calculate the square root of the right side: 0.594777072

Break this problem into two subproblems by setting 
(y + 0.796875) equal to 0.594777072 and -0.594777072.


Subproblem 1
y + 0.796875 = 0.594777072

Simplifying
y + 0.796875 = 0.594777072

Reorder the terms:
0.796875 + y = 0.594777072

Solving
0.796875 + y = 0.594777072

Solving for variable 'y'.

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

Add '-0.796875' to each side of the equation.
0.796875 + -0.796875 + y = 0.594777072 + -0.796875

Combine like terms: 0.796875 + -0.796875 = 0.000000
0.000000 + y = 0.594777072 + -0.796875
y = 0.594777072 + -0.796875

Combine like terms: 0.594777072 + -0.796875 = -0.202097928
y = -0.202097928

Simplifying
y = -0.202097928


Subproblem 2
y + 0.796875 = -0.594777072

Simplifying
y + 0.796875 = -0.594777072

Reorder the terms:
0.796875 + y = -0.594777072

Solving
0.796875 + y = -0.594777072

Solving for variable 'y'.

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

Add '-0.796875' to each side of the equation.
0.796875 + -0.796875 + y = -0.594777072 + -0.796875

Combine like terms: 0.796875 + -0.796875 = 0.000000
0.000000 + y = -0.594777072 + -0.796875
y = -0.594777072 + -0.796875

Combine like terms: -0.594777072 + -0.796875 = -1.391652072
y = -1.391652072

Simplifying
y = -1.391652072


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {-0.202097928, -1.391652072}
Solution
y = {-0.202097928, -1.391652072}

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