n^2-18n-105=0

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Solution for n^2-18n-105=0 equation:


Simplifying
n2 + -18n + -105 = 0

Reorder the terms:
-105 + -18n + n2 = 0

Solving
-105 + -18n + n2 = 0

Solving for variable 'n'.

Begin completing the square.

Move the constant term to the right:

Add '105' to each side of the equation.
-105 + -18n + 105 + n2 = 0 + 105

Reorder the terms:
-105 + 105 + -18n + n2 = 0 + 105

Combine like terms: -105 + 105 = 0
0 + -18n + n2 = 0 + 105
-18n + n2 = 0 + 105

Combine like terms: 0 + 105 = 105
-18n + n2 = 105

The n term is -18n.  Take half its coefficient (-9).
Square it (81) and add it to both sides.

Add '81' to each side of the equation.
-18n + 81 + n2 = 105 + 81

Reorder the terms:
81 + -18n + n2 = 105 + 81

Combine like terms: 105 + 81 = 186
81 + -18n + n2 = 186

Factor a perfect square on the left side:
(n + -9)(n + -9) = 186

Calculate the square root of the right side: 13.638181697

Break this problem into two subproblems by setting 
(n + -9) equal to 13.638181697 and -13.638181697.

Subproblem 1

n + -9 = 13.638181697 Simplifying n + -9 = 13.638181697 Reorder the terms: -9 + n = 13.638181697 Solving -9 + n = 13.638181697 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '9' to each side of the equation. -9 + 9 + n = 13.638181697 + 9 Combine like terms: -9 + 9 = 0 0 + n = 13.638181697 + 9 n = 13.638181697 + 9 Combine like terms: 13.638181697 + 9 = 22.638181697 n = 22.638181697 Simplifying n = 22.638181697

Subproblem 2

n + -9 = -13.638181697 Simplifying n + -9 = -13.638181697 Reorder the terms: -9 + n = -13.638181697 Solving -9 + n = -13.638181697 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '9' to each side of the equation. -9 + 9 + n = -13.638181697 + 9 Combine like terms: -9 + 9 = 0 0 + n = -13.638181697 + 9 n = -13.638181697 + 9 Combine like terms: -13.638181697 + 9 = -4.638181697 n = -4.638181697 Simplifying n = -4.638181697

Solution

The solution to the problem is based on the solutions from the subproblems. n = {22.638181697, -4.638181697}

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