v(11+v)=10

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Solution for v(11+v)=10 equation: 


Simplifying
v(11 + v) = 10
(11 * v + v * v) = 10
(11v + v2) = 10

Solving
11v + v2 = 10

Solving for variable 'v'.

Reorder the terms:
-10 + 11v + v2 = 10 + -10

Combine like terms: 10 + -10 = 0
-10 + 11v + v2 = 0

Begin completing the square.

Move the constant term to the right:

Add '10' to each side of the equation.
-10 + 11v + 10 + v2 = 0 + 10

Reorder the terms:
-10 + 10 + 11v + v2 = 0 + 10

Combine like terms: -10 + 10 = 0
0 + 11v + v2 = 0 + 10
11v + v2 = 0 + 10

Combine like terms: 0 + 10 = 10
11v + v2 = 10

The v term is 11v.  Take half its coefficient (5.5).
Square it (30.25) and add it to both sides.

Add '30.25' to each side of the equation.
11v + 30.25 + v2 = 10 + 30.25

Reorder the terms:
30.25 + 11v + v2 = 10 + 30.25

Combine like terms: 10 + 30.25 = 40.25
30.25 + 11v + v2 = 40.25

Factor a perfect square on the left side:
(v + 5.5)(v + 5.5) = 40.25

Calculate the square root of the right side: 6.34428877

Break this problem into two subproblems by setting 
(v + 5.5) equal to 6.34428877 and -6.34428877.


Subproblem 1
v + 5.5 = 6.34428877

Simplifying
v + 5.5 = 6.34428877

Reorder the terms:
5.5 + v = 6.34428877

Solving
5.5 + v = 6.34428877

Solving for variable 'v'.

Move all terms containing v to the left, all other terms to the right.

Add '-5.5' to each side of the equation.
5.5 + -5.5 + v = 6.34428877 + -5.5

Combine like terms: 5.5 + -5.5 = 0.0
0.0 + v = 6.34428877 + -5.5
v = 6.34428877 + -5.5

Combine like terms: 6.34428877 + -5.5 = 0.84428877
v = 0.84428877

Simplifying
v = 0.84428877


Subproblem 2
v + 5.5 = -6.34428877

Simplifying
v + 5.5 = -6.34428877

Reorder the terms:
5.5 + v = -6.34428877

Solving
5.5 + v = -6.34428877

Solving for variable 'v'.

Move all terms containing v to the left, all other terms to the right.

Add '-5.5' to each side of the equation.
5.5 + -5.5 + v = -6.34428877 + -5.5

Combine like terms: 5.5 + -5.5 = 0.0
0.0 + v = -6.34428877 + -5.5
v = -6.34428877 + -5.5

Combine like terms: -6.34428877 + -5.5 = -11.84428877
v = -11.84428877

Simplifying
v = -11.84428877


Solution
The solution to the problem is based on the solutions
from the subproblems.
v = {0.84428877, -11.84428877}

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