3v^2=v+12

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Solution for 3v^2=v+12 equation:


Simplifying
3v2 = v + 12

Reorder the terms:
3v2 = 12 + v

Solving
3v2 = 12 + v

Solving for variable 'v'.

Reorder the terms:
-12 + -1v + 3v2 = 12 + v + -12 + -1v

Reorder the terms:
-12 + -1v + 3v2 = 12 + -12 + v + -1v

Combine like terms: 12 + -12 = 0
-12 + -1v + 3v2 = 0 + v + -1v
-12 + -1v + 3v2 = v + -1v

Combine like terms: v + -1v = 0
-12 + -1v + 3v2 = 0

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

Divide each side by '3'.
-4 + -0.3333333333v + v2 = 0

Move the constant term to the right:

Add '4' to each side of the equation.
-4 + -0.3333333333v + 4 + v2 = 0 + 4

Reorder the terms:
-4 + 4 + -0.3333333333v + v2 = 0 + 4

Combine like terms: -4 + 4 = 0
0 + -0.3333333333v + v2 = 0 + 4
-0.3333333333v + v2 = 0 + 4

Combine like terms: 0 + 4 = 4
-0.3333333333v + v2 = 4

The v term is -0.3333333333v.  Take half its coefficient (-0.1666666667).
Square it (0.02777777779) and add it to both sides.

Add '0.02777777779' to each side of the equation.
-0.3333333333v + 0.02777777779 + v2 = 4 + 0.02777777779

Reorder the terms:
0.02777777779 + -0.3333333333v + v2 = 4 + 0.02777777779

Combine like terms: 4 + 0.02777777779 = 4.02777777779
0.02777777779 + -0.3333333333v + v2 = 4.02777777779

Factor a perfect square on the left side:
(v + -0.1666666667)(v + -0.1666666667) = 4.02777777779

Calculate the square root of the right side: 2.00693243

Break this problem into two subproblems by setting 
(v + -0.1666666667) equal to 2.00693243 and -2.00693243.

Subproblem 1

v + -0.1666666667 = 2.00693243 Simplifying v + -0.1666666667 = 2.00693243 Reorder the terms: -0.1666666667 + v = 2.00693243 Solving -0.1666666667 + v = 2.00693243 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '0.1666666667' to each side of the equation. -0.1666666667 + 0.1666666667 + v = 2.00693243 + 0.1666666667 Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000 0.0000000000 + v = 2.00693243 + 0.1666666667 v = 2.00693243 + 0.1666666667 Combine like terms: 2.00693243 + 0.1666666667 = 2.1735990967 v = 2.1735990967 Simplifying v = 2.1735990967

Subproblem 2

v + -0.1666666667 = -2.00693243 Simplifying v + -0.1666666667 = -2.00693243 Reorder the terms: -0.1666666667 + v = -2.00693243 Solving -0.1666666667 + v = -2.00693243 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '0.1666666667' to each side of the equation. -0.1666666667 + 0.1666666667 + v = -2.00693243 + 0.1666666667 Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000 0.0000000000 + v = -2.00693243 + 0.1666666667 v = -2.00693243 + 0.1666666667 Combine like terms: -2.00693243 + 0.1666666667 = -1.8402657633 v = -1.8402657633 Simplifying v = -1.8402657633

Solution

The solution to the problem is based on the solutions from the subproblems. v = {2.1735990967, -1.8402657633}

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