N^2-10N-41=0

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Solution for N^2-10N-41=0 equation: 


Simplifying
N2 + -10N + -41 = 0

Reorder the terms:
-41 + -10N + N2 = 0

Solving
-41 + -10N + N2 = 0

Solving for variable 'N'.

Begin completing the square.

Move the constant term to the right:

Add '41' to each side of the equation.
-41 + -10N + 41 + N2 = 0 + 41

Reorder the terms:
-41 + 41 + -10N + N2 = 0 + 41

Combine like terms: -41 + 41 = 0
0 + -10N + N2 = 0 + 41
-10N + N2 = 0 + 41

Combine like terms: 0 + 41 = 41
-10N + N2 = 41

The N term is -10N.  Take half its coefficient (-5).
Square it (25) and add it to both sides.

Add '25' to each side of the equation.
-10N + 25 + N2 = 41 + 25

Reorder the terms:
25 + -10N + N2 = 41 + 25

Combine like terms: 41 + 25 = 66
25 + -10N + N2 = 66

Factor a perfect square on the left side:
(N + -5)(N + -5) = 66

Calculate the square root of the right side: 8.124038405

Break this problem into two subproblems by setting 
(N + -5) equal to 8.124038405 and -8.124038405.


Subproblem 1
N + -5 = 8.124038405

Simplifying
N + -5 = 8.124038405

Reorder the terms:
-5 + N = 8.124038405

Solving
-5 + N = 8.124038405

Solving for variable 'N'.

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

Add '5' to each side of the equation.
-5 + 5 + N = 8.124038405 + 5

Combine like terms: -5 + 5 = 0
0 + N = 8.124038405 + 5
N = 8.124038405 + 5

Combine like terms: 8.124038405 + 5 = 13.124038405
N = 13.124038405

Simplifying
N = 13.124038405


Subproblem 2
N + -5 = -8.124038405

Simplifying
N + -5 = -8.124038405

Reorder the terms:
-5 + N = -8.124038405

Solving
-5 + N = -8.124038405

Solving for variable 'N'.

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

Add '5' to each side of the equation.
-5 + 5 + N = -8.124038405 + 5

Combine like terms: -5 + 5 = 0
0 + N = -8.124038405 + 5
N = -8.124038405 + 5

Combine like terms: -8.124038405 + 5 = -3.124038405
N = -3.124038405

Simplifying
N = -3.124038405


Solution
The solution to the problem is based on the solutions
from the subproblems.
N = {13.124038405, -3.124038405}

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