v^2-10v+23=0

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Solution for v^2-10v+23=0 equation: 


Simplifying
v2 + -10v + 23 = 0

Reorder the terms:
23 + -10v + v2 = 0

Solving
23 + -10v + v2 = 0

Solving for variable 'v'.

Begin completing the square.

Move the constant term to the right:

Add '-23' to each side of the equation.
23 + -10v + -23 + v2 = 0 + -23

Reorder the terms:
23 + -23 + -10v + v2 = 0 + -23

Combine like terms: 23 + -23 = 0
0 + -10v + v2 = 0 + -23
-10v + v2 = 0 + -23

Combine like terms: 0 + -23 = -23
-10v + v2 = -23

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 = -23 + 25

Reorder the terms:
25 + -10v + v2 = -23 + 25

Combine like terms: -23 + 25 = 2
25 + -10v + v2 = 2

Factor a perfect square on the left side:
(v + -5)(v + -5) = 2

Calculate the square root of the right side: 1.414213562

Break this problem into two subproblems by setting 
(v + -5) equal to 1.414213562 and -1.414213562.


Subproblem 1
v + -5 = 1.414213562

Simplifying
v + -5 = 1.414213562

Reorder the terms:
-5 + v = 1.414213562

Solving
-5 + v = 1.414213562

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 = 1.414213562 + 5

Combine like terms: -5 + 5 = 0
0 + v = 1.414213562 + 5
v = 1.414213562 + 5

Combine like terms: 1.414213562 + 5 = 6.414213562
v = 6.414213562

Simplifying
v = 6.414213562


Subproblem 2
v + -5 = -1.414213562

Simplifying
v + -5 = -1.414213562

Reorder the terms:
-5 + v = -1.414213562

Solving
-5 + v = -1.414213562

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 = -1.414213562 + 5

Combine like terms: -5 + 5 = 0
0 + v = -1.414213562 + 5
v = -1.414213562 + 5

Combine like terms: -1.414213562 + 5 = 3.585786438
v = 3.585786438

Simplifying
v = 3.585786438


Solution
The solution to the problem is based on the solutions
from the subproblems.
v = {6.414213562, 3.585786438}

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