8v^2-10v-2=0

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Solution for 8v^2-10v-2=0 equation: 


Simplifying
8v2 + -10v + -2 = 0

Reorder the terms:
-2 + -10v + 8v2 = 0

Solving
-2 + -10v + 8v2 = 0

Solving for variable 'v'.

Factor out the Greatest Common Factor (GCF), '2'.
2(-1 + -5v + 4v2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-1 + -5v + 4v2)' equal to zero and attempt to solve:

Simplifying
-1 + -5v + 4v2 = 0

Solving
-1 + -5v + 4v2 = 0

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

Divide each side by '4'.
-0.25 + -1.25v + v2 = 0

Move the constant term to the right:

Add '0.25' to each side of the equation.
-0.25 + -1.25v + 0.25 + v2 = 0 + 0.25

Reorder the terms:
-0.25 + 0.25 + -1.25v + v2 = 0 + 0.25

Combine like terms: -0.25 + 0.25 = 0.00
0.00 + -1.25v + v2 = 0 + 0.25
-1.25v + v2 = 0 + 0.25

Combine like terms: 0 + 0.25 = 0.25
-1.25v + v2 = 0.25

The v term is -1.25v.  Take half its coefficient (-0.625).
Square it (0.390625) and add it to both sides.

Add '0.390625' to each side of the equation.
-1.25v + 0.390625 + v2 = 0.25 + 0.390625

Reorder the terms:
0.390625 + -1.25v + v2 = 0.25 + 0.390625

Combine like terms: 0.25 + 0.390625 = 0.640625
0.390625 + -1.25v + v2 = 0.640625

Factor a perfect square on the left side:
(v + -0.625)(v + -0.625) = 0.640625

Calculate the square root of the right side: 0.80039053

Break this problem into two subproblems by setting 
(v + -0.625) equal to 0.80039053 and -0.80039053.


Subproblem 1
v + -0.625 = 0.80039053

Simplifying
v + -0.625 = 0.80039053

Reorder the terms:
-0.625 + v = 0.80039053

Solving
-0.625 + v = 0.80039053

Solving for variable 'v'.

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

Add '0.625' to each side of the equation.
-0.625 + 0.625 + v = 0.80039053 + 0.625

Combine like terms: -0.625 + 0.625 = 0.000
0.000 + v = 0.80039053 + 0.625
v = 0.80039053 + 0.625

Combine like terms: 0.80039053 + 0.625 = 1.42539053
v = 1.42539053

Simplifying
v = 1.42539053


Subproblem 2
v + -0.625 = -0.80039053

Simplifying
v + -0.625 = -0.80039053

Reorder the terms:
-0.625 + v = -0.80039053

Solving
-0.625 + v = -0.80039053

Solving for variable 'v'.

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

Add '0.625' to each side of the equation.
-0.625 + 0.625 + v = -0.80039053 + 0.625

Combine like terms: -0.625 + 0.625 = 0.000
0.000 + v = -0.80039053 + 0.625
v = -0.80039053 + 0.625

Combine like terms: -0.80039053 + 0.625 = -0.17539053
v = -0.17539053

Simplifying
v = -0.17539053


Solution
The solution to the problem is based on the solutions
from the subproblems.
v = {1.42539053, -0.17539053}
Solution
v = {1.42539053, -0.17539053}

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