v^2=10v+19

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


Simplifying
v2 = 10v + 19

Reorder the terms:
v2 = 19 + 10v

Solving
v2 = 19 + 10v

Solving for variable 'v'.

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

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

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

Combine like terms: 10v + -10v = 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 = 11.633249581 v = 11.633249581 Simplifying v = 11.633249581

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 = -1.633249581 v = -1.633249581 Simplifying v = -1.633249581

Solution

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

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