n^2-10n-116=-9

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Solution for n^2-10n-116=-9 equation:


Simplifying
n2 + -10n + -116 = -9

Reorder the terms:
-116 + -10n + n2 = -9

Solving
-116 + -10n + n2 = -9

Solving for variable 'n'.

Reorder the terms:
-116 + 9 + -10n + n2 = -9 + 9

Combine like terms: -116 + 9 = -107
-107 + -10n + n2 = -9 + 9

Combine like terms: -9 + 9 = 0
-107 + -10n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '107' to each side of the equation.
-107 + -10n + 107 + n2 = 0 + 107

Reorder the terms:
-107 + 107 + -10n + n2 = 0 + 107

Combine like terms: -107 + 107 = 0
0 + -10n + n2 = 0 + 107
-10n + n2 = 0 + 107

Combine like terms: 0 + 107 = 107
-10n + n2 = 107

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 = 107 + 25

Reorder the terms:
25 + -10n + n2 = 107 + 25

Combine like terms: 107 + 25 = 132
25 + -10n + n2 = 132

Factor a perfect square on the left side:
(n + -5)(n + -5) = 132

Calculate the square root of the right side: 11.489125293

Break this problem into two subproblems by setting 
(n + -5) equal to 11.489125293 and -11.489125293.

Subproblem 1

n + -5 = 11.489125293 Simplifying n + -5 = 11.489125293 Reorder the terms: -5 + n = 11.489125293 Solving -5 + n = 11.489125293 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 = 11.489125293 + 5 Combine like terms: -5 + 5 = 0 0 + n = 11.489125293 + 5 n = 11.489125293 + 5 Combine like terms: 11.489125293 + 5 = 16.489125293 n = 16.489125293 Simplifying n = 16.489125293

Subproblem 2

n + -5 = -11.489125293 Simplifying n + -5 = -11.489125293 Reorder the terms: -5 + n = -11.489125293 Solving -5 + n = -11.489125293 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 = -11.489125293 + 5 Combine like terms: -5 + 5 = 0 0 + n = -11.489125293 + 5 n = -11.489125293 + 5 Combine like terms: -11.489125293 + 5 = -6.489125293 n = -6.489125293 Simplifying n = -6.489125293

Solution

The solution to the problem is based on the solutions from the subproblems. n = {16.489125293, -6.489125293}

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