v^2+9v-106=0

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Solution for v^2+9v-106=0 equation:


Simplifying
v2 + 9v + -106 = 0

Reorder the terms:
-106 + 9v + v2 = 0

Solving
-106 + 9v + v2 = 0

Solving for variable 'v'.

Begin completing the square.

Move the constant term to the right:

Add '106' to each side of the equation.
-106 + 9v + 106 + v2 = 0 + 106

Reorder the terms:
-106 + 106 + 9v + v2 = 0 + 106

Combine like terms: -106 + 106 = 0
0 + 9v + v2 = 0 + 106
9v + v2 = 0 + 106

Combine like terms: 0 + 106 = 106
9v + v2 = 106

The v term is 9v.  Take half its coefficient (4.5).
Square it (20.25) and add it to both sides.

Add '20.25' to each side of the equation.
9v + 20.25 + v2 = 106 + 20.25

Reorder the terms:
20.25 + 9v + v2 = 106 + 20.25

Combine like terms: 106 + 20.25 = 126.25
20.25 + 9v + v2 = 126.25

Factor a perfect square on the left side:
(v + 4.5)(v + 4.5) = 126.25

Calculate the square root of the right side: 11.236102527

Break this problem into two subproblems by setting 
(v + 4.5) equal to 11.236102527 and -11.236102527.

Subproblem 1

v + 4.5 = 11.236102527 Simplifying v + 4.5 = 11.236102527 Reorder the terms: 4.5 + v = 11.236102527 Solving 4.5 + v = 11.236102527 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-4.5' to each side of the equation. 4.5 + -4.5 + v = 11.236102527 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + v = 11.236102527 + -4.5 v = 11.236102527 + -4.5 Combine like terms: 11.236102527 + -4.5 = 6.736102527 v = 6.736102527 Simplifying v = 6.736102527

Subproblem 2

v + 4.5 = -11.236102527 Simplifying v + 4.5 = -11.236102527 Reorder the terms: 4.5 + v = -11.236102527 Solving 4.5 + v = -11.236102527 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-4.5' to each side of the equation. 4.5 + -4.5 + v = -11.236102527 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + v = -11.236102527 + -4.5 v = -11.236102527 + -4.5 Combine like terms: -11.236102527 + -4.5 = -15.736102527 v = -15.736102527 Simplifying v = -15.736102527

Solution

The solution to the problem is based on the solutions from the subproblems. v = {6.736102527, -15.736102527}

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