n^2-9n-54=0

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Solution for n^2-9n-54=0 equation: 


Simplifying
n2 + -9n + -54 = 0

Reorder the terms:
-54 + -9n + n2 = 0

Solving
-54 + -9n + n2 = 0

Solving for variable 'n'.

Begin completing the square.

Move the constant term to the right:

Add '54' to each side of the equation.
-54 + -9n + 54 + n2 = 0 + 54

Reorder the terms:
-54 + 54 + -9n + n2 = 0 + 54

Combine like terms: -54 + 54 = 0
0 + -9n + n2 = 0 + 54
-9n + n2 = 0 + 54

Combine like terms: 0 + 54 = 54
-9n + n2 = 54

The n term is -9n.  Take half its coefficient (-4.5).
Square it (20.25) and add it to both sides.

Add '20.25' to each side of the equation.
-9n + 20.25 + n2 = 54 + 20.25

Reorder the terms:
20.25 + -9n + n2 = 54 + 20.25

Combine like terms: 54 + 20.25 = 74.25
20.25 + -9n + n2 = 74.25

Factor a perfect square on the left side:
(n + -4.5)(n + -4.5) = 74.25

Calculate the square root of the right side: 8.61684397

Break this problem into two subproblems by setting 
(n + -4.5) equal to 8.61684397 and -8.61684397.


Subproblem 1
n + -4.5 = 8.61684397

Simplifying
n + -4.5 = 8.61684397

Reorder the terms:
-4.5 + n = 8.61684397

Solving
-4.5 + n = 8.61684397

Solving for variable 'n'.

Move all terms containing n to the left, all other terms to the right.

Add '4.5' to each side of the equation.
-4.5 + 4.5 + n = 8.61684397 + 4.5

Combine like terms: -4.5 + 4.5 = 0.0
0.0 + n = 8.61684397 + 4.5
n = 8.61684397 + 4.5

Combine like terms: 8.61684397 + 4.5 = 13.11684397
n = 13.11684397

Simplifying
n = 13.11684397


Subproblem 2
n + -4.5 = -8.61684397

Simplifying
n + -4.5 = -8.61684397

Reorder the terms:
-4.5 + n = -8.61684397

Solving
-4.5 + n = -8.61684397

Solving for variable 'n'.

Move all terms containing n to the left, all other terms to the right.

Add '4.5' to each side of the equation.
-4.5 + 4.5 + n = -8.61684397 + 4.5

Combine like terms: -4.5 + 4.5 = 0.0
0.0 + n = -8.61684397 + 4.5
n = -8.61684397 + 4.5

Combine like terms: -8.61684397 + 4.5 = -4.11684397
n = -4.11684397

Simplifying
n = -4.11684397


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {13.11684397, -4.11684397}

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