n^2-5n-82=0

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Solution for n^2-5n-82=0 equation:


Simplifying
n2 + -5n + -82 = 0

Reorder the terms:
-82 + -5n + n2 = 0

Solving
-82 + -5n + n2 = 0

Solving for variable 'n'.

Begin completing the square.

Move the constant term to the right:

Add '82' to each side of the equation.
-82 + -5n + 82 + n2 = 0 + 82

Reorder the terms:
-82 + 82 + -5n + n2 = 0 + 82

Combine like terms: -82 + 82 = 0
0 + -5n + n2 = 0 + 82
-5n + n2 = 0 + 82

Combine like terms: 0 + 82 = 82
-5n + n2 = 82

The n term is -5n.  Take half its coefficient (-2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
-5n + 6.25 + n2 = 82 + 6.25

Reorder the terms:
6.25 + -5n + n2 = 82 + 6.25

Combine like terms: 82 + 6.25 = 88.25
6.25 + -5n + n2 = 88.25

Factor a perfect square on the left side:
(n + -2.5)(n + -2.5) = 88.25

Calculate the square root of the right side: 9.394147114

Break this problem into two subproblems by setting 
(n + -2.5) equal to 9.394147114 and -9.394147114.

Subproblem 1

n + -2.5 = 9.394147114 Simplifying n + -2.5 = 9.394147114 Reorder the terms: -2.5 + n = 9.394147114 Solving -2.5 + n = 9.394147114 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + n = 9.394147114 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + n = 9.394147114 + 2.5 n = 9.394147114 + 2.5 Combine like terms: 9.394147114 + 2.5 = 11.894147114 n = 11.894147114 Simplifying n = 11.894147114

Subproblem 2

n + -2.5 = -9.394147114 Simplifying n + -2.5 = -9.394147114 Reorder the terms: -2.5 + n = -9.394147114 Solving -2.5 + n = -9.394147114 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + n = -9.394147114 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + n = -9.394147114 + 2.5 n = -9.394147114 + 2.5 Combine like terms: -9.394147114 + 2.5 = -6.894147114 n = -6.894147114 Simplifying n = -6.894147114

Solution

The solution to the problem is based on the solutions from the subproblems. n = {11.894147114, -6.894147114}

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