8v^2+2v-19=0

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Solution for 8v^2+2v-19=0 equation: 


Simplifying
8v2 + 2v + -19 = 0

Reorder the terms:
-19 + 2v + 8v2 = 0

Solving
-19 + 2v + 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'.
-2.375 + 0.25v + v2 = 0

Move the constant term to the right:

Add '2.375' to each side of the equation.
-2.375 + 0.25v + 2.375 + v2 = 0 + 2.375

Reorder the terms:
-2.375 + 2.375 + 0.25v + v2 = 0 + 2.375

Combine like terms: -2.375 + 2.375 = 0.000
0.000 + 0.25v + v2 = 0 + 2.375
0.25v + v2 = 0 + 2.375

Combine like terms: 0 + 2.375 = 2.375
0.25v + v2 = 2.375

The v term is 0.25v.  Take half its coefficient (0.125).
Square it (0.015625) and add it to both sides.

Add '0.015625' to each side of the equation.
0.25v + 0.015625 + v2 = 2.375 + 0.015625

Reorder the terms:
0.015625 + 0.25v + v2 = 2.375 + 0.015625

Combine like terms: 2.375 + 0.015625 = 2.390625
0.015625 + 0.25v + v2 = 2.390625

Factor a perfect square on the left side:
(v + 0.125)(v + 0.125) = 2.390625

Calculate the square root of the right side: 1.54616461

Break this problem into two subproblems by setting 
(v + 0.125) equal to 1.54616461 and -1.54616461.


Subproblem 1
v + 0.125 = 1.54616461

Simplifying
v + 0.125 = 1.54616461

Reorder the terms:
0.125 + v = 1.54616461

Solving
0.125 + v = 1.54616461

Solving for variable 'v'.

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

Add '-0.125' to each side of the equation.
0.125 + -0.125 + v = 1.54616461 + -0.125

Combine like terms: 0.125 + -0.125 = 0.000
0.000 + v = 1.54616461 + -0.125
v = 1.54616461 + -0.125

Combine like terms: 1.54616461 + -0.125 = 1.42116461
v = 1.42116461

Simplifying
v = 1.42116461


Subproblem 2
v + 0.125 = -1.54616461

Simplifying
v + 0.125 = -1.54616461

Reorder the terms:
0.125 + v = -1.54616461

Solving
0.125 + v = -1.54616461

Solving for variable 'v'.

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

Add '-0.125' to each side of the equation.
0.125 + -0.125 + v = -1.54616461 + -0.125

Combine like terms: 0.125 + -0.125 = 0.000
0.000 + v = -1.54616461 + -0.125
v = -1.54616461 + -0.125

Combine like terms: -1.54616461 + -0.125 = -1.67116461
v = -1.67116461

Simplifying
v = -1.67116461


Solution
The solution to the problem is based on the solutions
from the subproblems.
v = {1.42116461, -1.67116461}

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