9v^2+1=12v

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


Simplifying
9v2 + 1 = 12v

Reorder the terms:
1 + 9v2 = 12v

Solving
1 + 9v2 = 12v

Solving for variable 'v'.

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

Combine like terms: 12v + -12v = 0
1 + -12v + 9v2 = 0

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 = 0.3333333331
0.4444444442 + -1.333333333v + v2 = 0.3333333331

Factor a perfect square on the left side:
(v + -0.6666666665)(v + -0.6666666665) = 0.3333333331

Calculate the square root of the right side: 0.577350269

Break this problem into two subproblems by setting 
(v + -0.6666666665) equal to 0.577350269 and -0.577350269.


Subproblem 1
v + -0.6666666665 = 0.577350269

Simplifying
v + -0.6666666665 = 0.577350269

Reorder the terms:
-0.6666666665 + v = 0.577350269

Solving
-0.6666666665 + v = 0.577350269

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 = 0.577350269 + 0.6666666665

Combine like terms: -0.6666666665 + 0.6666666665 = 0.0000000000
0.0000000000 + v = 0.577350269 + 0.6666666665
v = 0.577350269 + 0.6666666665

Combine like terms: 0.577350269 + 0.6666666665 = 1.2440169355
v = 1.2440169355

Simplifying
v = 1.2440169355


Subproblem 2
v + -0.6666666665 = -0.577350269

Simplifying
v + -0.6666666665 = -0.577350269

Reorder the terms:
-0.6666666665 + v = -0.577350269

Solving
-0.6666666665 + v = -0.577350269

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 = -0.577350269 + 0.6666666665

Combine like terms: -0.6666666665 + 0.6666666665 = 0.0000000000
0.0000000000 + v = -0.577350269 + 0.6666666665
v = -0.577350269 + 0.6666666665

Combine like terms: -0.577350269 + 0.6666666665 = 0.0893163975
v = 0.0893163975

Simplifying
v = 0.0893163975


Solution
The solution to the problem is based on the solutions
from the subproblems.
v = {1.2440169355, 0.0893163975}

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