6v^2=2v^2+16v+31

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Solution for 6v^2=2v^2+16v+31 equation: 


Simplifying
6v2 = 2v2 + 16v + 31

Reorder the terms:
6v2 = 31 + 16v + 2v2

Solving
6v2 = 31 + 16v + 2v2

Solving for variable 'v'.

Reorder the terms:
-31 + -16v + 6v2 + -2v2 = 31 + 16v + 2v2 + -31 + -16v + -2v2

Combine like terms: 6v2 + -2v2 = 4v2
-31 + -16v + 4v2 = 31 + 16v + 2v2 + -31 + -16v + -2v2

Reorder the terms:
-31 + -16v + 4v2 = 31 + -31 + 16v + -16v + 2v2 + -2v2

Combine like terms: 31 + -31 = 0
-31 + -16v + 4v2 = 0 + 16v + -16v + 2v2 + -2v2
-31 + -16v + 4v2 = 16v + -16v + 2v2 + -2v2

Combine like terms: 16v + -16v = 0
-31 + -16v + 4v2 = 0 + 2v2 + -2v2
-31 + -16v + 4v2 = 2v2 + -2v2

Combine like terms: 2v2 + -2v2 = 0
-31 + -16v + 4v2 = 0

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

Divide each side by '4'.
-7.75 + -4v + v2 = 0

Move the constant term to the right:

Add '7.75' to each side of the equation.
-7.75 + -4v + 7.75 + v2 = 0 + 7.75

Reorder the terms:
-7.75 + 7.75 + -4v + v2 = 0 + 7.75

Combine like terms: -7.75 + 7.75 = 0.00
0.00 + -4v + v2 = 0 + 7.75
-4v + v2 = 0 + 7.75

Combine like terms: 0 + 7.75 = 7.75
-4v + v2 = 7.75

The v term is -4v.  Take half its coefficient (-2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
-4v + 4 + v2 = 7.75 + 4

Reorder the terms:
4 + -4v + v2 = 7.75 + 4

Combine like terms: 7.75 + 4 = 11.75
4 + -4v + v2 = 11.75

Factor a perfect square on the left side:
(v + -2)(v + -2) = 11.75

Calculate the square root of the right side: 3.4278273

Break this problem into two subproblems by setting 
(v + -2) equal to 3.4278273 and -3.4278273.


Subproblem 1
v + -2 = 3.4278273

Simplifying
v + -2 = 3.4278273

Reorder the terms:
-2 + v = 3.4278273

Solving
-2 + v = 3.4278273

Solving for variable 'v'.

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

Add '2' to each side of the equation.
-2 + 2 + v = 3.4278273 + 2

Combine like terms: -2 + 2 = 0
0 + v = 3.4278273 + 2
v = 3.4278273 + 2

Combine like terms: 3.4278273 + 2 = 5.4278273
v = 5.4278273

Simplifying
v = 5.4278273


Subproblem 2
v + -2 = -3.4278273

Simplifying
v + -2 = -3.4278273

Reorder the terms:
-2 + v = -3.4278273

Solving
-2 + v = -3.4278273

Solving for variable 'v'.

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

Add '2' to each side of the equation.
-2 + 2 + v = -3.4278273 + 2

Combine like terms: -2 + 2 = 0
0 + v = -3.4278273 + 2
v = -3.4278273 + 2

Combine like terms: -3.4278273 + 2 = -1.4278273
v = -1.4278273

Simplifying
v = -1.4278273


Solution
The solution to the problem is based on the solutions
from the subproblems.
v = {5.4278273, -1.4278273}

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