n^2-41n+105=0

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Solution for n^2-41n+105=0 equation: 


Simplifying
n2 + -41n + 105 = 0

Reorder the terms:
105 + -41n + n2 = 0

Solving
105 + -41n + n2 = 0

Solving for variable 'n'.

Begin completing the square.

Move the constant term to the right:

Add '-105' to each side of the equation.
105 + -41n + -105 + n2 = 0 + -105

Reorder the terms:
105 + -105 + -41n + n2 = 0 + -105

Combine like terms: 105 + -105 = 0
0 + -41n + n2 = 0 + -105
-41n + n2 = 0 + -105

Combine like terms: 0 + -105 = -105
-41n + n2 = -105

The n term is -41n.  Take half its coefficient (-20.5).
Square it (420.25) and add it to both sides.

Add '420.25' to each side of the equation.
-41n + 420.25 + n2 = -105 + 420.25

Reorder the terms:
420.25 + -41n + n2 = -105 + 420.25

Combine like terms: -105 + 420.25 = 315.25
420.25 + -41n + n2 = 315.25

Factor a perfect square on the left side:
(n + -20.5)(n + -20.5) = 315.25

Calculate the square root of the right side: 17.755280905

Break this problem into two subproblems by setting 
(n + -20.5) equal to 17.755280905 and -17.755280905.


Subproblem 1
n + -20.5 = 17.755280905

Simplifying
n + -20.5 = 17.755280905

Reorder the terms:
-20.5 + n = 17.755280905

Solving
-20.5 + n = 17.755280905

Solving for variable 'n'.

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

Add '20.5' to each side of the equation.
-20.5 + 20.5 + n = 17.755280905 + 20.5

Combine like terms: -20.5 + 20.5 = 0.0
0.0 + n = 17.755280905 + 20.5
n = 17.755280905 + 20.5

Combine like terms: 17.755280905 + 20.5 = 38.255280905
n = 38.255280905

Simplifying
n = 38.255280905


Subproblem 2
n + -20.5 = -17.755280905

Simplifying
n + -20.5 = -17.755280905

Reorder the terms:
-20.5 + n = -17.755280905

Solving
-20.5 + n = -17.755280905

Solving for variable 'n'.

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

Add '20.5' to each side of the equation.
-20.5 + 20.5 + n = -17.755280905 + 20.5

Combine like terms: -20.5 + 20.5 = 0.0
0.0 + n = -17.755280905 + 20.5
n = -17.755280905 + 20.5

Combine like terms: -17.755280905 + 20.5 = 2.744719095
n = 2.744719095

Simplifying
n = 2.744719095


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {38.255280905, 2.744719095}

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