9v^2-2v-1=0

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Solution for 9v^2-2v-1=0 equation: 


Simplifying
9v2 + -2v + -1 = 0

Reorder the terms:
-1 + -2v + 9v2 = 0

Solving
-1 + -2v + 9v2 = 0

Solving for variable 'v'.

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

Divide each side by '9'.
-0.1111111111 + -0.2222222222v + v2 = 0

Move the constant term to the right:

Add '0.1111111111' to each side of the equation.
-0.1111111111 + -0.2222222222v + 0.1111111111 + v2 = 0 + 0.1111111111

Reorder the terms:
-0.1111111111 + 0.1111111111 + -0.2222222222v + v2 = 0 + 0.1111111111

Combine like terms: -0.1111111111 + 0.1111111111 = 0.0000000000
0.0000000000 + -0.2222222222v + v2 = 0 + 0.1111111111
-0.2222222222v + v2 = 0 + 0.1111111111

Combine like terms: 0 + 0.1111111111 = 0.1111111111
-0.2222222222v + v2 = 0.1111111111

The v term is -0.2222222222v.  Take half its coefficient (-0.1111111111).
Square it (0.01234567901) and add it to both sides.

Add '0.01234567901' to each side of the equation.
-0.2222222222v + 0.01234567901 + v2 = 0.1111111111 + 0.01234567901

Reorder the terms:
0.01234567901 + -0.2222222222v + v2 = 0.1111111111 + 0.01234567901

Combine like terms: 0.1111111111 + 0.01234567901 = 0.12345679011
0.01234567901 + -0.2222222222v + v2 = 0.12345679011

Factor a perfect square on the left side:
(v + -0.1111111111)(v + -0.1111111111) = 0.12345679011

Calculate the square root of the right side: 0.351364184

Break this problem into two subproblems by setting 
(v + -0.1111111111) equal to 0.351364184 and -0.351364184.


Subproblem 1
v + -0.1111111111 = 0.351364184

Simplifying
v + -0.1111111111 = 0.351364184

Reorder the terms:
-0.1111111111 + v = 0.351364184

Solving
-0.1111111111 + v = 0.351364184

Solving for variable 'v'.

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

Add '0.1111111111' to each side of the equation.
-0.1111111111 + 0.1111111111 + v = 0.351364184 + 0.1111111111

Combine like terms: -0.1111111111 + 0.1111111111 = 0.0000000000
0.0000000000 + v = 0.351364184 + 0.1111111111
v = 0.351364184 + 0.1111111111

Combine like terms: 0.351364184 + 0.1111111111 = 0.4624752951
v = 0.4624752951

Simplifying
v = 0.4624752951


Subproblem 2
v + -0.1111111111 = -0.351364184

Simplifying
v + -0.1111111111 = -0.351364184

Reorder the terms:
-0.1111111111 + v = -0.351364184

Solving
-0.1111111111 + v = -0.351364184

Solving for variable 'v'.

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

Add '0.1111111111' to each side of the equation.
-0.1111111111 + 0.1111111111 + v = -0.351364184 + 0.1111111111

Combine like terms: -0.1111111111 + 0.1111111111 = 0.0000000000
0.0000000000 + v = -0.351364184 + 0.1111111111
v = -0.351364184 + 0.1111111111

Combine like terms: -0.351364184 + 0.1111111111 = -0.2402530729
v = -0.2402530729

Simplifying
v = -0.2402530729


Solution
The solution to the problem is based on the solutions
from the subproblems.
v = {0.4624752951, -0.2402530729}

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