n^2+16n=117

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Solution for n^2+16n=117 equation:


Simplifying
n2 + 16n = 117

Reorder the terms:
16n + n2 = 117

Solving
16n + n2 = 117

Solving for variable 'n'.

Reorder the terms:
-117 + 16n + n2 = 117 + -117

Combine like terms: 117 + -117 = 0
-117 + 16n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '117' to each side of the equation.
-117 + 16n + 117 + n2 = 0 + 117

Reorder the terms:
-117 + 117 + 16n + n2 = 0 + 117

Combine like terms: -117 + 117 = 0
0 + 16n + n2 = 0 + 117
16n + n2 = 0 + 117

Combine like terms: 0 + 117 = 117
16n + n2 = 117

The n term is 16n.  Take half its coefficient (8).
Square it (64) and add it to both sides.

Add '64' to each side of the equation.
16n + 64 + n2 = 117 + 64

Reorder the terms:
64 + 16n + n2 = 117 + 64

Combine like terms: 117 + 64 = 181
64 + 16n + n2 = 181

Factor a perfect square on the left side:
(n + 8)(n + 8) = 181

Calculate the square root of the right side: 13.453624047

Break this problem into two subproblems by setting 
(n + 8) equal to 13.453624047 and -13.453624047.

Subproblem 1

n + 8 = 13.453624047 Simplifying n + 8 = 13.453624047 Reorder the terms: 8 + n = 13.453624047 Solving 8 + n = 13.453624047 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-8' to each side of the equation. 8 + -8 + n = 13.453624047 + -8 Combine like terms: 8 + -8 = 0 0 + n = 13.453624047 + -8 n = 13.453624047 + -8 Combine like terms: 13.453624047 + -8 = 5.453624047 n = 5.453624047 Simplifying n = 5.453624047

Subproblem 2

n + 8 = -13.453624047 Simplifying n + 8 = -13.453624047 Reorder the terms: 8 + n = -13.453624047 Solving 8 + n = -13.453624047 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-8' to each side of the equation. 8 + -8 + n = -13.453624047 + -8 Combine like terms: 8 + -8 = 0 0 + n = -13.453624047 + -8 n = -13.453624047 + -8 Combine like terms: -13.453624047 + -8 = -21.453624047 n = -21.453624047 Simplifying n = -21.453624047

Solution

The solution to the problem is based on the solutions from the subproblems. n = {5.453624047, -21.453624047}

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