n^2-81=-5n

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Solution for n^2-81=-5n equation:


Simplifying
n2 + -81 = -5n

Reorder the terms:
-81 + n2 = -5n

Solving
-81 + n2 = -5n

Solving for variable 'n'.

Reorder the terms:
-81 + 5n + n2 = -5n + 5n

Combine like terms: -5n + 5n = 0
-81 + 5n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '81' to each side of the equation.
-81 + 5n + 81 + n2 = 0 + 81

Reorder the terms:
-81 + 81 + 5n + n2 = 0 + 81

Combine like terms: -81 + 81 = 0
0 + 5n + n2 = 0 + 81
5n + n2 = 0 + 81

Combine like terms: 0 + 81 = 81
5n + n2 = 81

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 = 81 + 6.25

Reorder the terms:
6.25 + 5n + n2 = 81 + 6.25

Combine like terms: 81 + 6.25 = 87.25
6.25 + 5n + n2 = 87.25

Factor a perfect square on the left side:
(n + 2.5)(n + 2.5) = 87.25

Calculate the square root of the right side: 9.340770846

Break this problem into two subproblems by setting 
(n + 2.5) equal to 9.340770846 and -9.340770846.

Subproblem 1

n + 2.5 = 9.340770846 Simplifying n + 2.5 = 9.340770846 Reorder the terms: 2.5 + n = 9.340770846 Solving 2.5 + n = 9.340770846 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 = 9.340770846 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + n = 9.340770846 + -2.5 n = 9.340770846 + -2.5 Combine like terms: 9.340770846 + -2.5 = 6.840770846 n = 6.840770846 Simplifying n = 6.840770846

Subproblem 2

n + 2.5 = -9.340770846 Simplifying n + 2.5 = -9.340770846 Reorder the terms: 2.5 + n = -9.340770846 Solving 2.5 + n = -9.340770846 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 = -9.340770846 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + n = -9.340770846 + -2.5 n = -9.340770846 + -2.5 Combine like terms: -9.340770846 + -2.5 = -11.840770846 n = -11.840770846 Simplifying n = -11.840770846

Solution

The solution to the problem is based on the solutions from the subproblems. n = {6.840770846, -11.840770846}

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