n^2-15n-81=0

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Solution for n^2-15n-81=0 equation:


Simplifying
n2 + -15n + -81 = 0

Reorder the terms:
-81 + -15n + n2 = 0

Solving
-81 + -15n + n2 = 0

Solving for variable 'n'.

Begin completing the square.

Move the constant term to the right:

Add '81' to each side of the equation.
-81 + -15n + 81 + n2 = 0 + 81

Reorder the terms:
-81 + 81 + -15n + n2 = 0 + 81

Combine like terms: -81 + 81 = 0
0 + -15n + n2 = 0 + 81
-15n + n2 = 0 + 81

Combine like terms: 0 + 81 = 81
-15n + n2 = 81

The n term is -15n.  Take half its coefficient (-7.5).
Square it (56.25) and add it to both sides.

Add '56.25' to each side of the equation.
-15n + 56.25 + n2 = 81 + 56.25

Reorder the terms:
56.25 + -15n + n2 = 81 + 56.25

Combine like terms: 81 + 56.25 = 137.25
56.25 + -15n + n2 = 137.25

Factor a perfect square on the left side:
(n + -7.5)(n + -7.5) = 137.25

Calculate the square root of the right side: 11.715374514

Break this problem into two subproblems by setting 
(n + -7.5) equal to 11.715374514 and -11.715374514.

Subproblem 1

n + -7.5 = 11.715374514 Simplifying n + -7.5 = 11.715374514 Reorder the terms: -7.5 + n = 11.715374514 Solving -7.5 + n = 11.715374514 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '7.5' to each side of the equation. -7.5 + 7.5 + n = 11.715374514 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + n = 11.715374514 + 7.5 n = 11.715374514 + 7.5 Combine like terms: 11.715374514 + 7.5 = 19.215374514 n = 19.215374514 Simplifying n = 19.215374514

Subproblem 2

n + -7.5 = -11.715374514 Simplifying n + -7.5 = -11.715374514 Reorder the terms: -7.5 + n = -11.715374514 Solving -7.5 + n = -11.715374514 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '7.5' to each side of the equation. -7.5 + 7.5 + n = -11.715374514 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + n = -11.715374514 + 7.5 n = -11.715374514 + 7.5 Combine like terms: -11.715374514 + 7.5 = -4.215374514 n = -4.215374514 Simplifying n = -4.215374514

Solution

The solution to the problem is based on the solutions from the subproblems. n = {19.215374514, -4.215374514}

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