7v^2-2v-1=0

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Solution for 7v^2-2v-1=0 equation: 


Simplifying
7v2 + -2v + -1 = 0

Reorder the terms:
-1 + -2v + 7v2 = 0

Solving
-1 + -2v + 7v2 = 0

Solving for variable 'v'.

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

Divide each side by '7'.
-0.1428571429 + -0.2857142857v + v2 = 0

Move the constant term to the right:

Add '0.1428571429' to each side of the equation.
-0.1428571429 + -0.2857142857v + 0.1428571429 + v2 = 0 + 0.1428571429

Reorder the terms:
-0.1428571429 + 0.1428571429 + -0.2857142857v + v2 = 0 + 0.1428571429

Combine like terms: -0.1428571429 + 0.1428571429 = 0.0000000000
0.0000000000 + -0.2857142857v + v2 = 0 + 0.1428571429
-0.2857142857v + v2 = 0 + 0.1428571429

Combine like terms: 0 + 0.1428571429 = 0.1428571429
-0.2857142857v + v2 = 0.1428571429

The v term is -0.2857142857v.  Take half its coefficient (-0.1428571429).
Square it (0.02040816328) and add it to both sides.

Add '0.02040816328' to each side of the equation.
-0.2857142857v + 0.02040816328 + v2 = 0.1428571429 + 0.02040816328

Reorder the terms:
0.02040816328 + -0.2857142857v + v2 = 0.1428571429 + 0.02040816328

Combine like terms: 0.1428571429 + 0.02040816328 = 0.16326530618
0.02040816328 + -0.2857142857v + v2 = 0.16326530618

Factor a perfect square on the left side:
(v + -0.1428571429)(v + -0.1428571429) = 0.16326530618

Calculate the square root of the right side: 0.404061018

Break this problem into two subproblems by setting 
(v + -0.1428571429) equal to 0.404061018 and -0.404061018.


Subproblem 1
v + -0.1428571429 = 0.404061018

Simplifying
v + -0.1428571429 = 0.404061018

Reorder the terms:
-0.1428571429 + v = 0.404061018

Solving
-0.1428571429 + v = 0.404061018

Solving for variable 'v'.

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

Add '0.1428571429' to each side of the equation.
-0.1428571429 + 0.1428571429 + v = 0.404061018 + 0.1428571429

Combine like terms: -0.1428571429 + 0.1428571429 = 0.0000000000
0.0000000000 + v = 0.404061018 + 0.1428571429
v = 0.404061018 + 0.1428571429

Combine like terms: 0.404061018 + 0.1428571429 = 0.5469181609
v = 0.5469181609

Simplifying
v = 0.5469181609


Subproblem 2
v + -0.1428571429 = -0.404061018

Simplifying
v + -0.1428571429 = -0.404061018

Reorder the terms:
-0.1428571429 + v = -0.404061018

Solving
-0.1428571429 + v = -0.404061018

Solving for variable 'v'.

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

Add '0.1428571429' to each side of the equation.
-0.1428571429 + 0.1428571429 + v = -0.404061018 + 0.1428571429

Combine like terms: -0.1428571429 + 0.1428571429 = 0.0000000000
0.0000000000 + v = -0.404061018 + 0.1428571429
v = -0.404061018 + 0.1428571429

Combine like terms: -0.404061018 + 0.1428571429 = -0.2612038751
v = -0.2612038751

Simplifying
v = -0.2612038751


Solution
The solution to the problem is based on the solutions
from the subproblems.
v = {0.5469181609, -0.2612038751}

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