9v^2+18v-65=0

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Solution for 9v^2+18v-65=0 equation:


Simplifying
9v2 + 18v + -65 = 0

Reorder the terms:
-65 + 18v + 9v2 = 0

Solving
-65 + 18v + 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'.
-7.222222222 + 2v + v2 = 0

Move the constant term to the right:

Add '7.222222222' to each side of the equation.
-7.222222222 + 2v + 7.222222222 + v2 = 0 + 7.222222222

Reorder the terms:
-7.222222222 + 7.222222222 + 2v + v2 = 0 + 7.222222222

Combine like terms: -7.222222222 + 7.222222222 = 0.000000000
0.000000000 + 2v + v2 = 0 + 7.222222222
2v + v2 = 0 + 7.222222222

Combine like terms: 0 + 7.222222222 = 7.222222222
2v + v2 = 7.222222222

The v term is 2v.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2v + 1 + v2 = 7.222222222 + 1

Reorder the terms:
1 + 2v + v2 = 7.222222222 + 1

Combine like terms: 7.222222222 + 1 = 8.222222222
1 + 2v + v2 = 8.222222222

Factor a perfect square on the left side:
(v + 1)(v + 1) = 8.222222222

Calculate the square root of the right side: 2.867441756

Break this problem into two subproblems by setting 
(v + 1) equal to 2.867441756 and -2.867441756.

Subproblem 1

v + 1 = 2.867441756 Simplifying v + 1 = 2.867441756 Reorder the terms: 1 + v = 2.867441756 Solving 1 + v = 2.867441756 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + v = 2.867441756 + -1 Combine like terms: 1 + -1 = 0 0 + v = 2.867441756 + -1 v = 2.867441756 + -1 Combine like terms: 2.867441756 + -1 = 1.867441756 v = 1.867441756 Simplifying v = 1.867441756

Subproblem 2

v + 1 = -2.867441756 Simplifying v + 1 = -2.867441756 Reorder the terms: 1 + v = -2.867441756 Solving 1 + v = -2.867441756 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + v = -2.867441756 + -1 Combine like terms: 1 + -1 = 0 0 + v = -2.867441756 + -1 v = -2.867441756 + -1 Combine like terms: -2.867441756 + -1 = -3.867441756 v = -3.867441756 Simplifying v = -3.867441756

Solution

The solution to the problem is based on the solutions from the subproblems. v = {1.867441756, -3.867441756}

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