n^2-24n+108-5=0

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Solution for n^2-24n+108-5=0 equation: 


Simplifying
n2 + -24n + 108 + -5 = 0

Reorder the terms:
108 + -5 + -24n + n2 = 0

Combine like terms: 108 + -5 = 103
103 + -24n + n2 = 0

Solving
103 + -24n + n2 = 0

Solving for variable 'n'.

Begin completing the square.

Move the constant term to the right:

Add '-103' to each side of the equation.
103 + -24n + -103 + n2 = 0 + -103

Reorder the terms:
103 + -103 + -24n + n2 = 0 + -103

Combine like terms: 103 + -103 = 0
0 + -24n + n2 = 0 + -103
-24n + n2 = 0 + -103

Combine like terms: 0 + -103 = -103
-24n + n2 = -103

The n term is -24n.  Take half its coefficient (-12).
Square it (144) and add it to both sides.

Add '144' to each side of the equation.
-24n + 144 + n2 = -103 + 144

Reorder the terms:
144 + -24n + n2 = -103 + 144

Combine like terms: -103 + 144 = 41
144 + -24n + n2 = 41

Factor a perfect square on the left side:
(n + -12)(n + -12) = 41

Calculate the square root of the right side: 6.403124237

Break this problem into two subproblems by setting 
(n + -12) equal to 6.403124237 and -6.403124237.


Subproblem 1
n + -12 = 6.403124237

Simplifying
n + -12 = 6.403124237

Reorder the terms:
-12 + n = 6.403124237

Solving
-12 + n = 6.403124237

Solving for variable 'n'.

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

Add '12' to each side of the equation.
-12 + 12 + n = 6.403124237 + 12

Combine like terms: -12 + 12 = 0
0 + n = 6.403124237 + 12
n = 6.403124237 + 12

Combine like terms: 6.403124237 + 12 = 18.403124237
n = 18.403124237

Simplifying
n = 18.403124237


Subproblem 2
n + -12 = -6.403124237

Simplifying
n + -12 = -6.403124237

Reorder the terms:
-12 + n = -6.403124237

Solving
-12 + n = -6.403124237

Solving for variable 'n'.

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

Add '12' to each side of the equation.
-12 + 12 + n = -6.403124237 + 12

Combine like terms: -12 + 12 = 0
0 + n = -6.403124237 + 12
n = -6.403124237 + 12

Combine like terms: -6.403124237 + 12 = 5.596875763
n = 5.596875763

Simplifying
n = 5.596875763


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {18.403124237, 5.596875763}

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