n^2=-54-18n

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Solution for n^2=-54-18n equation:


Simplifying
n2 = -54 + -18n

Solving
n2 = -54 + -18n

Solving for variable 'n'.

Reorder the terms:
54 + 18n + n2 = -54 + -18n + 54 + 18n

Reorder the terms:
54 + 18n + n2 = -54 + 54 + -18n + 18n

Combine like terms: -54 + 54 = 0
54 + 18n + n2 = 0 + -18n + 18n
54 + 18n + n2 = -18n + 18n

Combine like terms: -18n + 18n = 0
54 + 18n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-54' to each side of the equation.
54 + 18n + -54 + n2 = 0 + -54

Reorder the terms:
54 + -54 + 18n + n2 = 0 + -54

Combine like terms: 54 + -54 = 0
0 + 18n + n2 = 0 + -54
18n + n2 = 0 + -54

Combine like terms: 0 + -54 = -54
18n + n2 = -54

The n term is 18n.  Take half its coefficient (9).
Square it (81) and add it to both sides.

Add '81' to each side of the equation.
18n + 81 + n2 = -54 + 81

Reorder the terms:
81 + 18n + n2 = -54 + 81

Combine like terms: -54 + 81 = 27
81 + 18n + n2 = 27

Factor a perfect square on the left side:
(n + 9)(n + 9) = 27

Calculate the square root of the right side: 5.196152423

Break this problem into two subproblems by setting 
(n + 9) equal to 5.196152423 and -5.196152423.

Subproblem 1

n + 9 = 5.196152423 Simplifying n + 9 = 5.196152423 Reorder the terms: 9 + n = 5.196152423 Solving 9 + n = 5.196152423 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-9' to each side of the equation. 9 + -9 + n = 5.196152423 + -9 Combine like terms: 9 + -9 = 0 0 + n = 5.196152423 + -9 n = 5.196152423 + -9 Combine like terms: 5.196152423 + -9 = -3.803847577 n = -3.803847577 Simplifying n = -3.803847577

Subproblem 2

n + 9 = -5.196152423 Simplifying n + 9 = -5.196152423 Reorder the terms: 9 + n = -5.196152423 Solving 9 + n = -5.196152423 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-9' to each side of the equation. 9 + -9 + n = -5.196152423 + -9 Combine like terms: 9 + -9 = 0 0 + n = -5.196152423 + -9 n = -5.196152423 + -9 Combine like terms: -5.196152423 + -9 = -14.196152423 n = -14.196152423 Simplifying n = -14.196152423

Solution

The solution to the problem is based on the solutions from the subproblems. n = {-3.803847577, -14.196152423}

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