v^2+10v=19

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Solution for v^2+10v=19 equation:


Simplifying
v2 + 10v = 19

Reorder the terms:
10v + v2 = 19

Solving
10v + v2 = 19

Solving for variable 'v'.

Reorder the terms:
-19 + 10v + v2 = 19 + -19

Combine like terms: 19 + -19 = 0
-19 + 10v + v2 = 0

Begin completing the square.

Move the constant term to the right:

Add '19' to each side of the equation.
-19 + 10v + 19 + v2 = 0 + 19

Reorder the terms:
-19 + 19 + 10v + v2 = 0 + 19

Combine like terms: -19 + 19 = 0
0 + 10v + v2 = 0 + 19
10v + v2 = 0 + 19

Combine like terms: 0 + 19 = 19
10v + v2 = 19

The v term is 10v.  Take half its coefficient (5).
Square it (25) and add it to both sides.

Add '25' to each side of the equation.
10v + 25 + v2 = 19 + 25

Reorder the terms:
25 + 10v + v2 = 19 + 25

Combine like terms: 19 + 25 = 44
25 + 10v + v2 = 44

Factor a perfect square on the left side:
(v + 5)(v + 5) = 44

Calculate the square root of the right side: 6.633249581

Break this problem into two subproblems by setting 
(v + 5) equal to 6.633249581 and -6.633249581.


Subproblem 1
v + 5 = 6.633249581

Simplifying
v + 5 = 6.633249581

Reorder the terms:
5 + v = 6.633249581

Solving
5 + v = 6.633249581

Solving for variable 'v'.

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

Add '-5' to each side of the equation.
5 + -5 + v = 6.633249581 + -5

Combine like terms: 5 + -5 = 0
0 + v = 6.633249581 + -5
v = 6.633249581 + -5

Combine like terms: 6.633249581 + -5 = 1.633249581
v = 1.633249581

Simplifying
v = 1.633249581


Subproblem 2
v + 5 = -6.633249581

Simplifying
v + 5 = -6.633249581

Reorder the terms:
5 + v = -6.633249581

Solving
5 + v = -6.633249581

Solving for variable 'v'.

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

Add '-5' to each side of the equation.
5 + -5 + v = -6.633249581 + -5

Combine like terms: 5 + -5 = 0
0 + v = -6.633249581 + -5
v = -6.633249581 + -5

Combine like terms: -6.633249581 + -5 = -11.633249581
v = -11.633249581

Simplifying
v = -11.633249581


Solution
The solution to the problem is based on the solutions
from the subproblems.
v = {1.633249581, -11.633249581}

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