6v^2-3v-1=0

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Solution for 6v^2-3v-1=0 equation:


Simplifying
6v2 + -3v + -1 = 0

Reorder the terms:
-1 + -3v + 6v2 = 0

Solving
-1 + -3v + 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'.
-0.1666666667 + -0.5v + v2 = 0

Move the constant term to the right:

Add '0.1666666667' to each side of the equation.
-0.1666666667 + -0.5v + 0.1666666667 + v2 = 0 + 0.1666666667

Reorder the terms:
-0.1666666667 + 0.1666666667 + -0.5v + v2 = 0 + 0.1666666667

Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000
0.0000000000 + -0.5v + v2 = 0 + 0.1666666667
-0.5v + v2 = 0 + 0.1666666667

Combine like terms: 0 + 0.1666666667 = 0.1666666667
-0.5v + v2 = 0.1666666667

The v term is -0.5v.  Take half its coefficient (-0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
-0.5v + 0.0625 + v2 = 0.1666666667 + 0.0625

Reorder the terms:
0.0625 + -0.5v + v2 = 0.1666666667 + 0.0625

Combine like terms: 0.1666666667 + 0.0625 = 0.2291666667
0.0625 + -0.5v + v2 = 0.2291666667

Factor a perfect square on the left side:
(v + -0.25)(v + -0.25) = 0.2291666667

Calculate the square root of the right side: 0.478713554

Break this problem into two subproblems by setting 
(v + -0.25) equal to 0.478713554 and -0.478713554.

Subproblem 1

v + -0.25 = 0.478713554 Simplifying v + -0.25 = 0.478713554 Reorder the terms: -0.25 + v = 0.478713554 Solving -0.25 + v = 0.478713554 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + v = 0.478713554 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + v = 0.478713554 + 0.25 v = 0.478713554 + 0.25 Combine like terms: 0.478713554 + 0.25 = 0.728713554 v = 0.728713554 Simplifying v = 0.728713554

Subproblem 2

v + -0.25 = -0.478713554 Simplifying v + -0.25 = -0.478713554 Reorder the terms: -0.25 + v = -0.478713554 Solving -0.25 + v = -0.478713554 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + v = -0.478713554 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + v = -0.478713554 + 0.25 v = -0.478713554 + 0.25 Combine like terms: -0.478713554 + 0.25 = -0.228713554 v = -0.228713554 Simplifying v = -0.228713554

Solution

The solution to the problem is based on the solutions from the subproblems. v = {0.728713554, -0.228713554}

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