-2v(v+9)=5v+31

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Solution for -2v(v+9)=5v+31 equation: 


Simplifying
-2v(v + 9) = 5v + 31

Reorder the terms:
-2v(9 + v) = 5v + 31
(9 * -2v + v * -2v) = 5v + 31
(-18v + -2v2) = 5v + 31

Reorder the terms:
-18v + -2v2 = 31 + 5v

Solving
-18v + -2v2 = 31 + 5v

Solving for variable 'v'.

Reorder the terms:
-31 + -18v + -5v + -2v2 = 31 + 5v + -31 + -5v

Combine like terms: -18v + -5v = -23v
-31 + -23v + -2v2 = 31 + 5v + -31 + -5v

Reorder the terms:
-31 + -23v + -2v2 = 31 + -31 + 5v + -5v

Combine like terms: 31 + -31 = 0
-31 + -23v + -2v2 = 0 + 5v + -5v
-31 + -23v + -2v2 = 5v + -5v

Combine like terms: 5v + -5v = 0
-31 + -23v + -2v2 = 0

Factor out the Greatest Common Factor (GCF), '-1'.
-1(31 + 23v + 2v2) = 0

Ignore the factor -1.

Subproblem 1
Set the factor '(31 + 23v + 2v2)' equal to zero and attempt to solve:

Simplifying
31 + 23v + 2v2 = 0

Solving
31 + 23v + 2v2 = 0

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

Divide each side by '2'.
15.5 + 11.5v + v2 = 0

Move the constant term to the right:

Add '-15.5' to each side of the equation.
15.5 + 11.5v + -15.5 + v2 = 0 + -15.5

Reorder the terms:
15.5 + -15.5 + 11.5v + v2 = 0 + -15.5

Combine like terms: 15.5 + -15.5 = 0.0
0.0 + 11.5v + v2 = 0 + -15.5
11.5v + v2 = 0 + -15.5

Combine like terms: 0 + -15.5 = -15.5
11.5v + v2 = -15.5

The v term is 11.5v.  Take half its coefficient (5.75).
Square it (33.0625) and add it to both sides.

Add '33.0625' to each side of the equation.
11.5v + 33.0625 + v2 = -15.5 + 33.0625

Reorder the terms:
33.0625 + 11.5v + v2 = -15.5 + 33.0625

Combine like terms: -15.5 + 33.0625 = 17.5625
33.0625 + 11.5v + v2 = 17.5625

Factor a perfect square on the left side:
(v + 5.75)(v + 5.75) = 17.5625

Calculate the square root of the right side: 4.190763654

Break this problem into two subproblems by setting 
(v + 5.75) equal to 4.190763654 and -4.190763654.


Subproblem 1
v + 5.75 = 4.190763654

Simplifying
v + 5.75 = 4.190763654

Reorder the terms:
5.75 + v = 4.190763654

Solving
5.75 + v = 4.190763654

Solving for variable 'v'.

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

Add '-5.75' to each side of the equation.
5.75 + -5.75 + v = 4.190763654 + -5.75

Combine like terms: 5.75 + -5.75 = 0.00
0.00 + v = 4.190763654 + -5.75
v = 4.190763654 + -5.75

Combine like terms: 4.190763654 + -5.75 = -1.559236346
v = -1.559236346

Simplifying
v = -1.559236346


Subproblem 2
v + 5.75 = -4.190763654

Simplifying
v + 5.75 = -4.190763654

Reorder the terms:
5.75 + v = -4.190763654

Solving
5.75 + v = -4.190763654

Solving for variable 'v'.

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

Add '-5.75' to each side of the equation.
5.75 + -5.75 + v = -4.190763654 + -5.75

Combine like terms: 5.75 + -5.75 = 0.00
0.00 + v = -4.190763654 + -5.75
v = -4.190763654 + -5.75

Combine like terms: -4.190763654 + -5.75 = -9.940763654
v = -9.940763654

Simplifying
v = -9.940763654


Solution
The solution to the problem is based on the solutions
from the subproblems.
v = {-1.559236346, -9.940763654}
Solution
v = {-1.559236346, -9.940763654}

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