8v^2-10v+1=0

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Solution for 8v^2-10v+1=0 equation: 


Simplifying
8v2 + -10v + 1 = 0

Reorder the terms:
1 + -10v + 8v2 = 0

Solving
1 + -10v + 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 + -1.25v + v2 = 0

Move the constant term to the right:

Add '-0.125' to each side of the equation.
0.125 + -1.25v + -0.125 + v2 = 0 + -0.125

Reorder the terms:
0.125 + -0.125 + -1.25v + v2 = 0 + -0.125

Combine like terms: 0.125 + -0.125 = 0.000
0.000 + -1.25v + v2 = 0 + -0.125
-1.25v + v2 = 0 + -0.125

Combine like terms: 0 + -0.125 = -0.125
-1.25v + v2 = -0.125

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 = -0.125 + 0.390625

Reorder the terms:
0.390625 + -1.25v + v2 = -0.125 + 0.390625

Combine like terms: -0.125 + 0.390625 = 0.265625
0.390625 + -1.25v + v2 = 0.265625

Factor a perfect square on the left side:
(v + -0.625)(v + -0.625) = 0.265625

Calculate the square root of the right side: 0.515388203

Break this problem into two subproblems by setting 
(v + -0.625) equal to 0.515388203 and -0.515388203.


Subproblem 1
v + -0.625 = 0.515388203

Simplifying
v + -0.625 = 0.515388203

Reorder the terms:
-0.625 + v = 0.515388203

Solving
-0.625 + v = 0.515388203

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 = 0.515388203 + 0.625

Combine like terms: -0.625 + 0.625 = 0.000
0.000 + v = 0.515388203 + 0.625
v = 0.515388203 + 0.625

Combine like terms: 0.515388203 + 0.625 = 1.140388203
v = 1.140388203

Simplifying
v = 1.140388203


Subproblem 2
v + -0.625 = -0.515388203

Simplifying
v + -0.625 = -0.515388203

Reorder the terms:
-0.625 + v = -0.515388203

Solving
-0.625 + v = -0.515388203

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 = -0.515388203 + 0.625

Combine like terms: -0.625 + 0.625 = 0.000
0.000 + v = -0.515388203 + 0.625
v = -0.515388203 + 0.625

Combine like terms: -0.515388203 + 0.625 = 0.109611797
v = 0.109611797

Simplifying
v = 0.109611797


Solution
The solution to the problem is based on the solutions
from the subproblems.
v = {1.140388203, 0.109611797}

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