8v^2-6v-1=0

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Solution for 8v^2-6v-1=0 equation:


Simplifying
8v2 + -6v + -1 = 0

Reorder the terms:
-1 + -6v + 8v2 = 0

Solving
-1 + -6v + 8v2 = 0

Solving for variable 'v'.

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

Divide each side by '8'.
-0.125 + -0.75v + v2 = 0

Move the constant term to the right:

Add '0.125' to each side of the equation.
-0.125 + -0.75v + 0.125 + v2 = 0 + 0.125

Reorder the terms:
-0.125 + 0.125 + -0.75v + v2 = 0 + 0.125

Combine like terms: -0.125 + 0.125 = 0.000
0.000 + -0.75v + v2 = 0 + 0.125
-0.75v + v2 = 0 + 0.125

Combine like terms: 0 + 0.125 = 0.125
-0.75v + v2 = 0.125

The v term is -0.75v.  Take half its coefficient (-0.375).
Square it (0.140625) and add it to both sides.

Add '0.140625' to each side of the equation.
-0.75v + 0.140625 + v2 = 0.125 + 0.140625

Reorder the terms:
0.140625 + -0.75v + v2 = 0.125 + 0.140625

Combine like terms: 0.125 + 0.140625 = 0.265625
0.140625 + -0.75v + v2 = 0.265625

Factor a perfect square on the left side:
(v + -0.375)(v + -0.375) = 0.265625

Calculate the square root of the right side: 0.515388203

Break this problem into two subproblems by setting 
(v + -0.375) equal to 0.515388203 and -0.515388203.

Subproblem 1

v + -0.375 = 0.515388203 Simplifying v + -0.375 = 0.515388203 Reorder the terms: -0.375 + v = 0.515388203 Solving -0.375 + v = 0.515388203 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + v = 0.515388203 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + v = 0.515388203 + 0.375 v = 0.515388203 + 0.375 Combine like terms: 0.515388203 + 0.375 = 0.890388203 v = 0.890388203 Simplifying v = 0.890388203

Subproblem 2

v + -0.375 = -0.515388203 Simplifying v + -0.375 = -0.515388203 Reorder the terms: -0.375 + v = -0.515388203 Solving -0.375 + v = -0.515388203 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + v = -0.515388203 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + v = -0.515388203 + 0.375 v = -0.515388203 + 0.375 Combine like terms: -0.515388203 + 0.375 = -0.140388203 v = -0.140388203 Simplifying v = -0.140388203

Solution

The solution to the problem is based on the solutions from the subproblems. v = {0.890388203, -0.140388203}

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