3v^2-3v-23=0

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Solution for 3v^2-3v-23=0 equation: 


Simplifying
3v2 + -3v + -23 = 0

Reorder the terms:
-23 + -3v + 3v2 = 0

Solving
-23 + -3v + 3v2 = 0

Solving for variable 'v'.

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

Divide each side by '3'.
-7.666666667 + -1v + v2 = 0

Move the constant term to the right:

Add '7.666666667' to each side of the equation.
-7.666666667 + -1v + 7.666666667 + v2 = 0 + 7.666666667

Reorder the terms:
-7.666666667 + 7.666666667 + -1v + v2 = 0 + 7.666666667

Combine like terms: -7.666666667 + 7.666666667 = 0.000000000
0.000000000 + -1v + v2 = 0 + 7.666666667
-1v + v2 = 0 + 7.666666667

Combine like terms: 0 + 7.666666667 = 7.666666667
-1v + v2 = 7.666666667

The v term is -1v.  Take half its coefficient (-0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1v + 0.25 + v2 = 7.666666667 + 0.25

Reorder the terms:
0.25 + -1v + v2 = 7.666666667 + 0.25

Combine like terms: 7.666666667 + 0.25 = 7.916666667
0.25 + -1v + v2 = 7.916666667

Factor a perfect square on the left side:
(v + -0.5)(v + -0.5) = 7.916666667

Calculate the square root of the right side: 2.813657169

Break this problem into two subproblems by setting 
(v + -0.5) equal to 2.813657169 and -2.813657169.


Subproblem 1
v + -0.5 = 2.813657169

Simplifying
v + -0.5 = 2.813657169

Reorder the terms:
-0.5 + v = 2.813657169

Solving
-0.5 + v = 2.813657169

Solving for variable 'v'.

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

Add '0.5' to each side of the equation.
-0.5 + 0.5 + v = 2.813657169 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + v = 2.813657169 + 0.5
v = 2.813657169 + 0.5

Combine like terms: 2.813657169 + 0.5 = 3.313657169
v = 3.313657169

Simplifying
v = 3.313657169


Subproblem 2
v + -0.5 = -2.813657169

Simplifying
v + -0.5 = -2.813657169

Reorder the terms:
-0.5 + v = -2.813657169

Solving
-0.5 + v = -2.813657169

Solving for variable 'v'.

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

Add '0.5' to each side of the equation.
-0.5 + 0.5 + v = -2.813657169 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + v = -2.813657169 + 0.5
v = -2.813657169 + 0.5

Combine like terms: -2.813657169 + 0.5 = -2.313657169
v = -2.313657169

Simplifying
v = -2.313657169


Solution
The solution to the problem is based on the solutions
from the subproblems.
v = {3.313657169, -2.313657169}

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