9v^2+12v-1=0

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Solution for 9v^2+12v-1=0 equation: 


Simplifying
9v2 + 12v + -1 = 0

Reorder the terms:
-1 + 12v + 9v2 = 0

Solving
-1 + 12v + 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 + 1.333333333v + v2 = 0

Move the constant term to the right:

Add '0.1111111111' to each side of the equation.
-0.1111111111 + 1.333333333v + 0.1111111111 + v2 = 0 + 0.1111111111

Reorder the terms:
-0.1111111111 + 0.1111111111 + 1.333333333v + v2 = 0 + 0.1111111111

Combine like terms: -0.1111111111 + 0.1111111111 = 0.0000000000
0.0000000000 + 1.333333333v + v2 = 0 + 0.1111111111
1.333333333v + v2 = 0 + 0.1111111111

Combine like terms: 0 + 0.1111111111 = 0.1111111111
1.333333333v + v2 = 0.1111111111

The v term is 1.333333333v.  Take half its coefficient (0.6666666665).
Square it (0.4444444442) and add it to both sides.

Add '0.4444444442' to each side of the equation.
1.333333333v + 0.4444444442 + v2 = 0.1111111111 + 0.4444444442

Reorder the terms:
0.4444444442 + 1.333333333v + v2 = 0.1111111111 + 0.4444444442

Combine like terms: 0.1111111111 + 0.4444444442 = 0.5555555553
0.4444444442 + 1.333333333v + v2 = 0.5555555553

Factor a perfect square on the left side:
(v + 0.6666666665)(v + 0.6666666665) = 0.5555555553

Calculate the square root of the right side: 0.745355992

Break this problem into two subproblems by setting 
(v + 0.6666666665) equal to 0.745355992 and -0.745355992.


Subproblem 1
v + 0.6666666665 = 0.745355992

Simplifying
v + 0.6666666665 = 0.745355992

Reorder the terms:
0.6666666665 + v = 0.745355992

Solving
0.6666666665 + v = 0.745355992

Solving for variable 'v'.

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

Add '-0.6666666665' to each side of the equation.
0.6666666665 + -0.6666666665 + v = 0.745355992 + -0.6666666665

Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000
0.0000000000 + v = 0.745355992 + -0.6666666665
v = 0.745355992 + -0.6666666665

Combine like terms: 0.745355992 + -0.6666666665 = 0.0786893255
v = 0.0786893255

Simplifying
v = 0.0786893255


Subproblem 2
v + 0.6666666665 = -0.745355992

Simplifying
v + 0.6666666665 = -0.745355992

Reorder the terms:
0.6666666665 + v = -0.745355992

Solving
0.6666666665 + v = -0.745355992

Solving for variable 'v'.

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

Add '-0.6666666665' to each side of the equation.
0.6666666665 + -0.6666666665 + v = -0.745355992 + -0.6666666665

Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000
0.0000000000 + v = -0.745355992 + -0.6666666665
v = -0.745355992 + -0.6666666665

Combine like terms: -0.745355992 + -0.6666666665 = -1.4120226585
v = -1.4120226585

Simplifying
v = -1.4120226585


Solution
The solution to the problem is based on the solutions
from the subproblems.
v = {0.0786893255, -1.4120226585}

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