6Y^2+32Y-48=0

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Solution for 6Y^2+32Y-48=0 equation: 


Simplifying
6Y2 + 32Y + -48 = 0

Reorder the terms:
-48 + 32Y + 6Y2 = 0

Solving
-48 + 32Y + 6Y2 = 0

Solving for variable 'Y'.

Factor out the Greatest Common Factor (GCF), '2'.
2(-24 + 16Y + 3Y2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-24 + 16Y + 3Y2)' equal to zero and attempt to solve:

Simplifying
-24 + 16Y + 3Y2 = 0

Solving
-24 + 16Y + 3Y2 = 0

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

Divide each side by '3'.
-8 + 5.333333333Y + Y2 = 0

Move the constant term to the right:

Add '8' to each side of the equation.
-8 + 5.333333333Y + 8 + Y2 = 0 + 8

Reorder the terms:
-8 + 8 + 5.333333333Y + Y2 = 0 + 8

Combine like terms: -8 + 8 = 0
0 + 5.333333333Y + Y2 = 0 + 8
5.333333333Y + Y2 = 0 + 8

Combine like terms: 0 + 8 = 8
5.333333333Y + Y2 = 8

The Y term is 5.333333333Y.  Take half its coefficient (2.666666667).
Square it (7.111111113) and add it to both sides.

Add '7.111111113' to each side of the equation.
5.333333333Y + 7.111111113 + Y2 = 8 + 7.111111113

Reorder the terms:
7.111111113 + 5.333333333Y + Y2 = 8 + 7.111111113

Combine like terms: 8 + 7.111111113 = 15.111111113
7.111111113 + 5.333333333Y + Y2 = 15.111111113

Factor a perfect square on the left side:
(Y + 2.666666667)(Y + 2.666666667) = 15.111111113

Calculate the square root of the right side: 3.887301263

Break this problem into two subproblems by setting 
(Y + 2.666666667) equal to 3.887301263 and -3.887301263.


Subproblem 1
Y + 2.666666667 = 3.887301263

Simplifying
Y + 2.666666667 = 3.887301263

Reorder the terms:
2.666666667 + Y = 3.887301263

Solving
2.666666667 + Y = 3.887301263

Solving for variable 'Y'.

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

Add '-2.666666667' to each side of the equation.
2.666666667 + -2.666666667 + Y = 3.887301263 + -2.666666667

Combine like terms: 2.666666667 + -2.666666667 = 0.000000000
0.000000000 + Y = 3.887301263 + -2.666666667
Y = 3.887301263 + -2.666666667

Combine like terms: 3.887301263 + -2.666666667 = 1.220634596
Y = 1.220634596

Simplifying
Y = 1.220634596


Subproblem 2
Y + 2.666666667 = -3.887301263

Simplifying
Y + 2.666666667 = -3.887301263

Reorder the terms:
2.666666667 + Y = -3.887301263

Solving
2.666666667 + Y = -3.887301263

Solving for variable 'Y'.

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

Add '-2.666666667' to each side of the equation.
2.666666667 + -2.666666667 + Y = -3.887301263 + -2.666666667

Combine like terms: 2.666666667 + -2.666666667 = 0.000000000
0.000000000 + Y = -3.887301263 + -2.666666667
Y = -3.887301263 + -2.666666667

Combine like terms: -3.887301263 + -2.666666667 = -6.55396793
Y = -6.55396793

Simplifying
Y = -6.55396793


Solution
The solution to the problem is based on the solutions
from the subproblems.
Y = {1.220634596, -6.55396793}
Solution
Y = {1.220634596, -6.55396793}

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