4v^2-22v+5=0

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Solution for 4v^2-22v+5=0 equation: 


Simplifying
4v2 + -22v + 5 = 0

Reorder the terms:
5 + -22v + 4v2 = 0

Solving
5 + -22v + 4v2 = 0

Solving for variable 'v'.

Begin completing the square.  Divide all terms by
4 the coefficient of the squared term: 

Divide each side by '4'.
1.25 + -5.5v + v2 = 0

Move the constant term to the right:

Add '-1.25' to each side of the equation.
1.25 + -5.5v + -1.25 + v2 = 0 + -1.25

Reorder the terms:
1.25 + -1.25 + -5.5v + v2 = 0 + -1.25

Combine like terms: 1.25 + -1.25 = 0.00
0.00 + -5.5v + v2 = 0 + -1.25
-5.5v + v2 = 0 + -1.25

Combine like terms: 0 + -1.25 = -1.25
-5.5v + v2 = -1.25

The v term is -5.5v.  Take half its coefficient (-2.75).
Square it (7.5625) and add it to both sides.

Add '7.5625' to each side of the equation.
-5.5v + 7.5625 + v2 = -1.25 + 7.5625

Reorder the terms:
7.5625 + -5.5v + v2 = -1.25 + 7.5625

Combine like terms: -1.25 + 7.5625 = 6.3125
7.5625 + -5.5v + v2 = 6.3125

Factor a perfect square on the left side:
(v + -2.75)(v + -2.75) = 6.3125

Calculate the square root of the right side: 2.512468905

Break this problem into two subproblems by setting 
(v + -2.75) equal to 2.512468905 and -2.512468905.


Subproblem 1
v + -2.75 = 2.512468905

Simplifying
v + -2.75 = 2.512468905

Reorder the terms:
-2.75 + v = 2.512468905

Solving
-2.75 + v = 2.512468905

Solving for variable 'v'.

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

Add '2.75' to each side of the equation.
-2.75 + 2.75 + v = 2.512468905 + 2.75

Combine like terms: -2.75 + 2.75 = 0.00
0.00 + v = 2.512468905 + 2.75
v = 2.512468905 + 2.75

Combine like terms: 2.512468905 + 2.75 = 5.262468905
v = 5.262468905

Simplifying
v = 5.262468905


Subproblem 2
v + -2.75 = -2.512468905

Simplifying
v + -2.75 = -2.512468905

Reorder the terms:
-2.75 + v = -2.512468905

Solving
-2.75 + v = -2.512468905

Solving for variable 'v'.

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

Add '2.75' to each side of the equation.
-2.75 + 2.75 + v = -2.512468905 + 2.75

Combine like terms: -2.75 + 2.75 = 0.00
0.00 + v = -2.512468905 + 2.75
v = -2.512468905 + 2.75

Combine like terms: -2.512468905 + 2.75 = 0.237531095
v = 0.237531095

Simplifying
v = 0.237531095


Solution
The solution to the problem is based on the solutions
from the subproblems.
v = {5.262468905, 0.237531095}

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