8v^2-5v-1=0

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Solution for 8v^2-5v-1=0 equation: 


Simplifying
8v2 + -5v + -1 = 0

Reorder the terms:
-1 + -5v + 8v2 = 0

Solving
-1 + -5v + 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 + -0.625v + v2 = 0

Move the constant term to the right:

Add '0.125' to each side of the equation.
-0.125 + -0.625v + 0.125 + v2 = 0 + 0.125

Reorder the terms:
-0.125 + 0.125 + -0.625v + v2 = 0 + 0.125

Combine like terms: -0.125 + 0.125 = 0.000
0.000 + -0.625v + v2 = 0 + 0.125
-0.625v + v2 = 0 + 0.125

Combine like terms: 0 + 0.125 = 0.125
-0.625v + v2 = 0.125

The v term is -0.625v.  Take half its coefficient (-0.3125).
Square it (0.09765625) and add it to both sides.

Add '0.09765625' to each side of the equation.
-0.625v + 0.09765625 + v2 = 0.125 + 0.09765625

Reorder the terms:
0.09765625 + -0.625v + v2 = 0.125 + 0.09765625

Combine like terms: 0.125 + 0.09765625 = 0.22265625
0.09765625 + -0.625v + v2 = 0.22265625

Factor a perfect square on the left side:
(v + -0.3125)(v + -0.3125) = 0.22265625

Calculate the square root of the right side: 0.471864652

Break this problem into two subproblems by setting 
(v + -0.3125) equal to 0.471864652 and -0.471864652.


Subproblem 1
v + -0.3125 = 0.471864652

Simplifying
v + -0.3125 = 0.471864652

Reorder the terms:
-0.3125 + v = 0.471864652

Solving
-0.3125 + v = 0.471864652

Solving for variable 'v'.

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

Add '0.3125' to each side of the equation.
-0.3125 + 0.3125 + v = 0.471864652 + 0.3125

Combine like terms: -0.3125 + 0.3125 = 0.0000
0.0000 + v = 0.471864652 + 0.3125
v = 0.471864652 + 0.3125

Combine like terms: 0.471864652 + 0.3125 = 0.784364652
v = 0.784364652

Simplifying
v = 0.784364652


Subproblem 2
v + -0.3125 = -0.471864652

Simplifying
v + -0.3125 = -0.471864652

Reorder the terms:
-0.3125 + v = -0.471864652

Solving
-0.3125 + v = -0.471864652

Solving for variable 'v'.

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

Add '0.3125' to each side of the equation.
-0.3125 + 0.3125 + v = -0.471864652 + 0.3125

Combine like terms: -0.3125 + 0.3125 = 0.0000
0.0000 + v = -0.471864652 + 0.3125
v = -0.471864652 + 0.3125

Combine like terms: -0.471864652 + 0.3125 = -0.159364652
v = -0.159364652

Simplifying
v = -0.159364652


Solution
The solution to the problem is based on the solutions
from the subproblems.
v = {0.784364652, -0.159364652}

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