6v^2-5v-3=0

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Solution for 6v^2-5v-3=0 equation: 


Simplifying
6v2 + -5v + -3 = 0

Reorder the terms:
-3 + -5v + 6v2 = 0

Solving
-3 + -5v + 6v2 = 0

Solving for variable 'v'.

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

Divide each side by '6'.
-0.5 + -0.8333333333v + v2 = 0

Move the constant term to the right:

Add '0.5' to each side of the equation.
-0.5 + -0.8333333333v + 0.5 + v2 = 0 + 0.5

Reorder the terms:
-0.5 + 0.5 + -0.8333333333v + v2 = 0 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + -0.8333333333v + v2 = 0 + 0.5
-0.8333333333v + v2 = 0 + 0.5

Combine like terms: 0 + 0.5 = 0.5
-0.8333333333v + v2 = 0.5

The v term is -0.8333333333v.  Take half its coefficient (-0.4166666667).
Square it (0.1736111111) and add it to both sides.

Add '0.1736111111' to each side of the equation.
-0.8333333333v + 0.1736111111 + v2 = 0.5 + 0.1736111111

Reorder the terms:
0.1736111111 + -0.8333333333v + v2 = 0.5 + 0.1736111111

Combine like terms: 0.5 + 0.1736111111 = 0.6736111111
0.1736111111 + -0.8333333333v + v2 = 0.6736111111

Factor a perfect square on the left side:
(v + -0.4166666667)(v + -0.4166666667) = 0.6736111111

Calculate the square root of the right side: 0.82073815

Break this problem into two subproblems by setting 
(v + -0.4166666667) equal to 0.82073815 and -0.82073815.


Subproblem 1
v + -0.4166666667 = 0.82073815

Simplifying
v + -0.4166666667 = 0.82073815

Reorder the terms:
-0.4166666667 + v = 0.82073815

Solving
-0.4166666667 + v = 0.82073815

Solving for variable 'v'.

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

Add '0.4166666667' to each side of the equation.
-0.4166666667 + 0.4166666667 + v = 0.82073815 + 0.4166666667

Combine like terms: -0.4166666667 + 0.4166666667 = 0.0000000000
0.0000000000 + v = 0.82073815 + 0.4166666667
v = 0.82073815 + 0.4166666667

Combine like terms: 0.82073815 + 0.4166666667 = 1.2374048167
v = 1.2374048167

Simplifying
v = 1.2374048167


Subproblem 2
v + -0.4166666667 = -0.82073815

Simplifying
v + -0.4166666667 = -0.82073815

Reorder the terms:
-0.4166666667 + v = -0.82073815

Solving
-0.4166666667 + v = -0.82073815

Solving for variable 'v'.

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

Add '0.4166666667' to each side of the equation.
-0.4166666667 + 0.4166666667 + v = -0.82073815 + 0.4166666667

Combine like terms: -0.4166666667 + 0.4166666667 = 0.0000000000
0.0000000000 + v = -0.82073815 + 0.4166666667
v = -0.82073815 + 0.4166666667

Combine like terms: -0.82073815 + 0.4166666667 = -0.4040714833
v = -0.4040714833

Simplifying
v = -0.4040714833


Solution
The solution to the problem is based on the solutions
from the subproblems.
v = {1.2374048167, -0.4040714833}

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