v^2+5v-1=0

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Solution for v^2+5v-1=0 equation: 


Simplifying
v2 + 5v + -1 = 0

Reorder the terms:
-1 + 5v + v2 = 0

Solving
-1 + 5v + v2 = 0

Solving for variable 'v'.

Begin completing the square.

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + 5v + 1 + v2 = 0 + 1

Reorder the terms:
-1 + 1 + 5v + v2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + 5v + v2 = 0 + 1
5v + v2 = 0 + 1

Combine like terms: 0 + 1 = 1
5v + v2 = 1

The v term is 5v.  Take half its coefficient (2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
5v + 6.25 + v2 = 1 + 6.25

Reorder the terms:
6.25 + 5v + v2 = 1 + 6.25

Combine like terms: 1 + 6.25 = 7.25
6.25 + 5v + v2 = 7.25

Factor a perfect square on the left side:
(v + 2.5)(v + 2.5) = 7.25

Calculate the square root of the right side: 2.692582404

Break this problem into two subproblems by setting 
(v + 2.5) equal to 2.692582404 and -2.692582404.


Subproblem 1
v + 2.5 = 2.692582404

Simplifying
v + 2.5 = 2.692582404

Reorder the terms:
2.5 + v = 2.692582404

Solving
2.5 + v = 2.692582404

Solving for variable 'v'.

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

Add '-2.5' to each side of the equation.
2.5 + -2.5 + v = 2.692582404 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + v = 2.692582404 + -2.5
v = 2.692582404 + -2.5

Combine like terms: 2.692582404 + -2.5 = 0.192582404
v = 0.192582404

Simplifying
v = 0.192582404


Subproblem 2
v + 2.5 = -2.692582404

Simplifying
v + 2.5 = -2.692582404

Reorder the terms:
2.5 + v = -2.692582404

Solving
2.5 + v = -2.692582404

Solving for variable 'v'.

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

Add '-2.5' to each side of the equation.
2.5 + -2.5 + v = -2.692582404 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + v = -2.692582404 + -2.5
v = -2.692582404 + -2.5

Combine like terms: -2.692582404 + -2.5 = -5.192582404
v = -5.192582404

Simplifying
v = -5.192582404


Solution
The solution to the problem is based on the solutions
from the subproblems.
v = {0.192582404, -5.192582404}

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