(n+11)n=84

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Solution for (n+11)n=84 equation:


Simplifying
(n + 11) * n = 84

Reorder the terms:
(11 + n) * n = 84

Reorder the terms for easier multiplication:
n(11 + n) = 84
(11 * n + n * n) = 84
(11n + n2) = 84

Solving
11n + n2 = 84

Solving for variable 'n'.

Reorder the terms:
-84 + 11n + n2 = 84 + -84

Combine like terms: 84 + -84 = 0
-84 + 11n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '84' to each side of the equation.
-84 + 11n + 84 + n2 = 0 + 84

Reorder the terms:
-84 + 84 + 11n + n2 = 0 + 84

Combine like terms: -84 + 84 = 0
0 + 11n + n2 = 0 + 84
11n + n2 = 0 + 84

Combine like terms: 0 + 84 = 84
11n + n2 = 84

The n term is 11n.  Take half its coefficient (5.5).
Square it (30.25) and add it to both sides.

Add '30.25' to each side of the equation.
11n + 30.25 + n2 = 84 + 30.25

Reorder the terms:
30.25 + 11n + n2 = 84 + 30.25

Combine like terms: 84 + 30.25 = 114.25
30.25 + 11n + n2 = 114.25

Factor a perfect square on the left side:
(n + 5.5)(n + 5.5) = 114.25

Calculate the square root of the right side: 10.688779163

Break this problem into two subproblems by setting 
(n + 5.5) equal to 10.688779163 and -10.688779163.

Subproblem 1

n + 5.5 = 10.688779163 Simplifying n + 5.5 = 10.688779163 Reorder the terms: 5.5 + n = 10.688779163 Solving 5.5 + n = 10.688779163 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-5.5' to each side of the equation. 5.5 + -5.5 + n = 10.688779163 + -5.5 Combine like terms: 5.5 + -5.5 = 0.0 0.0 + n = 10.688779163 + -5.5 n = 10.688779163 + -5.5 Combine like terms: 10.688779163 + -5.5 = 5.188779163 n = 5.188779163 Simplifying n = 5.188779163

Subproblem 2

n + 5.5 = -10.688779163 Simplifying n + 5.5 = -10.688779163 Reorder the terms: 5.5 + n = -10.688779163 Solving 5.5 + n = -10.688779163 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-5.5' to each side of the equation. 5.5 + -5.5 + n = -10.688779163 + -5.5 Combine like terms: 5.5 + -5.5 = 0.0 0.0 + n = -10.688779163 + -5.5 n = -10.688779163 + -5.5 Combine like terms: -10.688779163 + -5.5 = -16.188779163 n = -16.188779163 Simplifying n = -16.188779163

Solution

The solution to the problem is based on the solutions from the subproblems. n = {5.188779163, -16.188779163}

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