3v^2+9v=-5

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Solution for 3v^2+9v=-5 equation:


Simplifying
3v2 + 9v = -5

Reorder the terms:
9v + 3v2 = -5

Solving
9v + 3v2 = -5

Solving for variable 'v'.

Reorder the terms:
5 + 9v + 3v2 = -5 + 5

Combine like terms: -5 + 5 = 0
5 + 9v + 3v2 = 0

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

Divide each side by '3'.
1.666666667 + 3v + v2 = 0

Move the constant term to the right:

Add '-1.666666667' to each side of the equation.
1.666666667 + 3v + -1.666666667 + v2 = 0 + -1.666666667

Reorder the terms:
1.666666667 + -1.666666667 + 3v + v2 = 0 + -1.666666667

Combine like terms: 1.666666667 + -1.666666667 = 0.000000000
0.000000000 + 3v + v2 = 0 + -1.666666667
3v + v2 = 0 + -1.666666667

Combine like terms: 0 + -1.666666667 = -1.666666667
3v + v2 = -1.666666667

The v term is 3v.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3v + 2.25 + v2 = -1.666666667 + 2.25

Reorder the terms:
2.25 + 3v + v2 = -1.666666667 + 2.25

Combine like terms: -1.666666667 + 2.25 = 0.583333333
2.25 + 3v + v2 = 0.583333333

Factor a perfect square on the left side:
(v + 1.5)(v + 1.5) = 0.583333333

Calculate the square root of the right side: 0.763762616

Break this problem into two subproblems by setting 
(v + 1.5) equal to 0.763762616 and -0.763762616.

Subproblem 1

v + 1.5 = 0.763762616 Simplifying v + 1.5 = 0.763762616 Reorder the terms: 1.5 + v = 0.763762616 Solving 1.5 + v = 0.763762616 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + v = 0.763762616 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + v = 0.763762616 + -1.5 v = 0.763762616 + -1.5 Combine like terms: 0.763762616 + -1.5 = -0.736237384 v = -0.736237384 Simplifying v = -0.736237384

Subproblem 2

v + 1.5 = -0.763762616 Simplifying v + 1.5 = -0.763762616 Reorder the terms: 1.5 + v = -0.763762616 Solving 1.5 + v = -0.763762616 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + v = -0.763762616 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + v = -0.763762616 + -1.5 v = -0.763762616 + -1.5 Combine like terms: -0.763762616 + -1.5 = -2.263762616 v = -2.263762616 Simplifying v = -2.263762616

Solution

The solution to the problem is based on the solutions from the subproblems. v = {-0.736237384, -2.263762616}

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