-v+3v^2=21

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Solution for -v+3v^2=21 equation: 


Simplifying
-1v + 3v2 = 21

Solving
-1v + 3v2 = 21

Solving for variable 'v'.

Reorder the terms:
-21 + -1v + 3v2 = 21 + -21

Combine like terms: 21 + -21 = 0
-21 + -1v + 3v2 = 0

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

Divide each side by '3'.
-7 + -0.3333333333v + v2 = 0

Move the constant term to the right:

Add '7' to each side of the equation.
-7 + -0.3333333333v + 7 + v2 = 0 + 7

Reorder the terms:
-7 + 7 + -0.3333333333v + v2 = 0 + 7

Combine like terms: -7 + 7 = 0
0 + -0.3333333333v + v2 = 0 + 7
-0.3333333333v + v2 = 0 + 7

Combine like terms: 0 + 7 = 7
-0.3333333333v + v2 = 7

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 = 7 + 0.02777777779

Reorder the terms:
0.02777777779 + -0.3333333333v + v2 = 7 + 0.02777777779

Combine like terms: 7 + 0.02777777779 = 7.02777777779
0.02777777779 + -0.3333333333v + v2 = 7.02777777779

Factor a perfect square on the left side:
(v + -0.1666666667)(v + -0.1666666667) = 7.02777777779

Calculate the square root of the right side: 2.65099562

Break this problem into two subproblems by setting 
(v + -0.1666666667) equal to 2.65099562 and -2.65099562.


Subproblem 1
v + -0.1666666667 = 2.65099562

Simplifying
v + -0.1666666667 = 2.65099562

Reorder the terms:
-0.1666666667 + v = 2.65099562

Solving
-0.1666666667 + v = 2.65099562

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 = 2.65099562 + 0.1666666667

Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000
0.0000000000 + v = 2.65099562 + 0.1666666667
v = 2.65099562 + 0.1666666667

Combine like terms: 2.65099562 + 0.1666666667 = 2.8176622867
v = 2.8176622867

Simplifying
v = 2.8176622867


Subproblem 2
v + -0.1666666667 = -2.65099562

Simplifying
v + -0.1666666667 = -2.65099562

Reorder the terms:
-0.1666666667 + v = -2.65099562

Solving
-0.1666666667 + v = -2.65099562

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 = -2.65099562 + 0.1666666667

Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000
0.0000000000 + v = -2.65099562 + 0.1666666667
v = -2.65099562 + 0.1666666667

Combine like terms: -2.65099562 + 0.1666666667 = -2.4843289533
v = -2.4843289533

Simplifying
v = -2.4843289533


Solution
The solution to the problem is based on the solutions
from the subproblems.
v = {2.8176622867, -2.4843289533}

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