-9v(v+4)+4v+6=6v+7

Solution for -9v(v+4)+4v+6=6v+7 equation:

Simplifying
-9v(v + 4) + 4v + 6 = 6v + 7

Reorder the terms:
-9v(4 + v) + 4v + 6 = 6v + 7
(4 * -9v + v * -9v) + 4v + 6 = 6v + 7
(-36v + -9v2) + 4v + 6 = 6v + 7

Reorder the terms:
6 + -36v + 4v + -9v2 = 6v + 7

Combine like terms: -36v + 4v = -32v
6 + -32v + -9v2 = 6v + 7

Reorder the terms:
6 + -32v + -9v2 = 7 + 6v

Solving
6 + -32v + -9v2 = 7 + 6v

Solving for variable 'v'.

Reorder the terms:
6 + -7 + -32v + -6v + -9v2 = 7 + 6v + -7 + -6v

Combine like terms: 6 + -7 = -1
-1 + -32v + -6v + -9v2 = 7 + 6v + -7 + -6v

Combine like terms: -32v + -6v = -38v
-1 + -38v + -9v2 = 7 + 6v + -7 + -6v

Reorder the terms:
-1 + -38v + -9v2 = 7 + -7 + 6v + -6v

Combine like terms: 7 + -7 = 0
-1 + -38v + -9v2 = 0 + 6v + -6v
-1 + -38v + -9v2 = 6v + -6v

Combine like terms: 6v + -6v = 0
-1 + -38v + -9v2 = 0

Factor out the Greatest Common Factor (GCF), '-1'.
-1(1 + 38v + 9v2) = 0

Ignore the factor -1.
Subproblem 1
Set the factor '(1 + 38v + 9v2)' equal to zero and attempt to solve:

Simplifying
1 + 38v + 9v2 = 0

Solving
1 + 38v + 9v2 = 0

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

Divide each side by '9'.
0.1111111111 + 4.222222222v + v2 = 0

Move the constant term to the right:

Add '-0.1111111111' to each side of the equation.
0.1111111111 + 4.222222222v + -0.1111111111 + v2 = 0 + -0.1111111111

Reorder the terms:
0.1111111111 + -0.1111111111 + 4.222222222v + v2 = 0 + -0.1111111111

Combine like terms: 0.1111111111 + -0.1111111111 = 0.0000000000
0.0000000000 + 4.222222222v + v2 = 0 + -0.1111111111
4.222222222v + v2 = 0 + -0.1111111111

Combine like terms: 0 + -0.1111111111 = -0.1111111111
4.222222222v + v2 = -0.1111111111

The v term is 4.222222222v.  Take half its coefficient (2.111111111).
Square it (4.456790123) and add it to both sides.

Add '4.456790123' to each side of the equation.
4.222222222v + 4.456790123 + v2 = -0.1111111111 + 4.456790123

Reorder the terms:
4.456790123 + 4.222222222v + v2 = -0.1111111111 + 4.456790123

Combine like terms: -0.1111111111 + 4.456790123 = 4.3456790119
4.456790123 + 4.222222222v + v2 = 4.3456790119

Factor a perfect square on the left side:
(v + 2.111111111)(v + 2.111111111) = 4.3456790119

Calculate the square root of the right side: 2.084629226

Break this problem into two subproblems by setting 
(v + 2.111111111) equal to 2.084629226 and -2.084629226.

Subproblem 1
v + 2.111111111 = 2.084629226

Simplifying
v + 2.111111111 = 2.084629226

Reorder the terms:
2.111111111 + v = 2.084629226

Solving
2.111111111 + v = 2.084629226

Solving for variable 'v'.

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

Add '-2.111111111' to each side of the equation.
2.111111111 + -2.111111111 + v = 2.084629226 + -2.111111111

Combine like terms: 2.111111111 + -2.111111111 = 0.000000000
0.000000000 + v = 2.084629226 + -2.111111111
v = 2.084629226 + -2.111111111

Combine like terms: 2.084629226 + -2.111111111 = -0.026481885
v = -0.026481885

Simplifying
v = -0.026481885

Subproblem 2
v + 2.111111111 = -2.084629226

Simplifying
v + 2.111111111 = -2.084629226

Reorder the terms:
2.111111111 + v = -2.084629226

Solving
2.111111111 + v = -2.084629226

Solving for variable 'v'.

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

Add '-2.111111111' to each side of the equation.
2.111111111 + -2.111111111 + v = -2.084629226 + -2.111111111

Combine like terms: 2.111111111 + -2.111111111 = 0.000000000
0.000000000 + v = -2.084629226 + -2.111111111
v = -2.084629226 + -2.111111111

Combine like terms: -2.084629226 + -2.111111111 = -4.195740337
v = -4.195740337

Simplifying
v = -4.195740337

Solution
The solution to the problem is based on the solutions
from the subproblems.
v = {-0.026481885, -4.195740337}
Solution
v = {-0.026481885, -4.195740337}