6y^2+27y+5=0

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Solution for 6y^2+27y+5=0 equation: 


Simplifying
6y2 + 27y + 5 = 0

Reorder the terms:
5 + 27y + 6y2 = 0

Solving
5 + 27y + 6y2 = 0

Solving for variable 'y'.

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

Divide each side by '6'.
0.8333333333 + 4.5y + y2 = 0

Move the constant term to the right:

Add '-0.8333333333' to each side of the equation.
0.8333333333 + 4.5y + -0.8333333333 + y2 = 0 + -0.8333333333

Reorder the terms:
0.8333333333 + -0.8333333333 + 4.5y + y2 = 0 + -0.8333333333

Combine like terms: 0.8333333333 + -0.8333333333 = 0.0000000000
0.0000000000 + 4.5y + y2 = 0 + -0.8333333333
4.5y + y2 = 0 + -0.8333333333

Combine like terms: 0 + -0.8333333333 = -0.8333333333
4.5y + y2 = -0.8333333333

The y term is 4.5y.  Take half its coefficient (2.25).
Square it (5.0625) and add it to both sides.

Add '5.0625' to each side of the equation.
4.5y + 5.0625 + y2 = -0.8333333333 + 5.0625

Reorder the terms:
5.0625 + 4.5y + y2 = -0.8333333333 + 5.0625

Combine like terms: -0.8333333333 + 5.0625 = 4.2291666667
5.0625 + 4.5y + y2 = 4.2291666667

Factor a perfect square on the left side:
(y + 2.25)(y + 2.25) = 4.2291666667

Calculate the square root of the right side: 2.05649378

Break this problem into two subproblems by setting 
(y + 2.25) equal to 2.05649378 and -2.05649378.


Subproblem 1
y + 2.25 = 2.05649378

Simplifying
y + 2.25 = 2.05649378

Reorder the terms:
2.25 + y = 2.05649378

Solving
2.25 + y = 2.05649378

Solving for variable 'y'.

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

Add '-2.25' to each side of the equation.
2.25 + -2.25 + y = 2.05649378 + -2.25

Combine like terms: 2.25 + -2.25 = 0.00
0.00 + y = 2.05649378 + -2.25
y = 2.05649378 + -2.25

Combine like terms: 2.05649378 + -2.25 = -0.19350622
y = -0.19350622

Simplifying
y = -0.19350622


Subproblem 2
y + 2.25 = -2.05649378

Simplifying
y + 2.25 = -2.05649378

Reorder the terms:
2.25 + y = -2.05649378

Solving
2.25 + y = -2.05649378

Solving for variable 'y'.

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

Add '-2.25' to each side of the equation.
2.25 + -2.25 + y = -2.05649378 + -2.25

Combine like terms: 2.25 + -2.25 = 0.00
0.00 + y = -2.05649378 + -2.25
y = -2.05649378 + -2.25

Combine like terms: -2.05649378 + -2.25 = -4.30649378
y = -4.30649378

Simplifying
y = -4.30649378


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {-0.19350622, -4.30649378}

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