6v^2+27v+15=0

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Solution for 6v^2+27v+15=0 equation: 


Simplifying
6v2 + 27v + 15 = 0

Reorder the terms:
15 + 27v + 6v2 = 0

Solving
15 + 27v + 6v2 = 0

Solving for variable 'v'.

Factor out the Greatest Common Factor (GCF), '3'.
3(5 + 9v + 2v2) = 0

Ignore the factor 3.

Subproblem 1
Set the factor '(5 + 9v + 2v2)' equal to zero and attempt to solve:

Simplifying
5 + 9v + 2v2 = 0

Solving
5 + 9v + 2v2 = 0

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

Divide each side by '2'.
2.5 + 4.5v + v2 = 0

Move the constant term to the right:

Add '-2.5' to each side of the equation.
2.5 + 4.5v + -2.5 + v2 = 0 + -2.5

Reorder the terms:
2.5 + -2.5 + 4.5v + v2 = 0 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + 4.5v + v2 = 0 + -2.5
4.5v + v2 = 0 + -2.5

Combine like terms: 0 + -2.5 = -2.5
4.5v + v2 = -2.5

The v term is 4.5v.  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.
4.5v + 5.0625 + v2 = -2.5 + 5.0625

Reorder the terms:
5.0625 + 4.5v + v2 = -2.5 + 5.0625

Combine like terms: -2.5 + 5.0625 = 2.5625
5.0625 + 4.5v + v2 = 2.5625

Factor a perfect square on the left side:
(v + 2.25)(v + 2.25) = 2.5625

Calculate the square root of the right side: 1.600781059

Break this problem into two subproblems by setting 
(v + 2.25) equal to 1.600781059 and -1.600781059.


Subproblem 1
v + 2.25 = 1.600781059

Simplifying
v + 2.25 = 1.600781059

Reorder the terms:
2.25 + v = 1.600781059

Solving
2.25 + v = 1.600781059

Solving for variable 'v'.

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

Add '-2.25' to each side of the equation.
2.25 + -2.25 + v = 1.600781059 + -2.25

Combine like terms: 2.25 + -2.25 = 0.00
0.00 + v = 1.600781059 + -2.25
v = 1.600781059 + -2.25

Combine like terms: 1.600781059 + -2.25 = -0.649218941
v = -0.649218941

Simplifying
v = -0.649218941


Subproblem 2
v + 2.25 = -1.600781059

Simplifying
v + 2.25 = -1.600781059

Reorder the terms:
2.25 + v = -1.600781059

Solving
2.25 + v = -1.600781059

Solving for variable 'v'.

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

Add '-2.25' to each side of the equation.
2.25 + -2.25 + v = -1.600781059 + -2.25

Combine like terms: 2.25 + -2.25 = 0.00
0.00 + v = -1.600781059 + -2.25
v = -1.600781059 + -2.25

Combine like terms: -1.600781059 + -2.25 = -3.850781059
v = -3.850781059

Simplifying
v = -3.850781059


Solution
The solution to the problem is based on the solutions
from the subproblems.
v = {-0.649218941, -3.850781059}
Solution
v = {-0.649218941, -3.850781059}

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