n^2+6n-108=0

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Solution for n^2+6n-108=0 equation:


Simplifying
n2 + 6n + -108 = 0

Reorder the terms:
-108 + 6n + n2 = 0

Solving
-108 + 6n + n2 = 0

Solving for variable 'n'.

Begin completing the square.

Move the constant term to the right:

Add '108' to each side of the equation.
-108 + 6n + 108 + n2 = 0 + 108

Reorder the terms:
-108 + 108 + 6n + n2 = 0 + 108

Combine like terms: -108 + 108 = 0
0 + 6n + n2 = 0 + 108
6n + n2 = 0 + 108

Combine like terms: 0 + 108 = 108
6n + n2 = 108

The n term is 6n.  Take half its coefficient (3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
6n + 9 + n2 = 108 + 9

Reorder the terms:
9 + 6n + n2 = 108 + 9

Combine like terms: 108 + 9 = 117
9 + 6n + n2 = 117

Factor a perfect square on the left side:
(n + 3)(n + 3) = 117

Calculate the square root of the right side: 10.816653826

Break this problem into two subproblems by setting 
(n + 3) equal to 10.816653826 and -10.816653826.

Subproblem 1

n + 3 = 10.816653826 Simplifying n + 3 = 10.816653826 Reorder the terms: 3 + n = 10.816653826 Solving 3 + n = 10.816653826 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + n = 10.816653826 + -3 Combine like terms: 3 + -3 = 0 0 + n = 10.816653826 + -3 n = 10.816653826 + -3 Combine like terms: 10.816653826 + -3 = 7.816653826 n = 7.816653826 Simplifying n = 7.816653826

Subproblem 2

n + 3 = -10.816653826 Simplifying n + 3 = -10.816653826 Reorder the terms: 3 + n = -10.816653826 Solving 3 + n = -10.816653826 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + n = -10.816653826 + -3 Combine like terms: 3 + -3 = 0 0 + n = -10.816653826 + -3 n = -10.816653826 + -3 Combine like terms: -10.816653826 + -3 = -13.816653826 n = -13.816653826 Simplifying n = -13.816653826

Solution

The solution to the problem is based on the solutions from the subproblems. n = {7.816653826, -13.816653826}

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