18n^2+11n-54=12n^2

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Solution for 18n^2+11n-54=12n^2 equation:


Simplifying
18n2 + 11n + -54 = 12n2

Reorder the terms:
-54 + 11n + 18n2 = 12n2

Solving
-54 + 11n + 18n2 = 12n2

Solving for variable 'n'.

Combine like terms: 18n2 + -12n2 = 6n2
-54 + 11n + 6n2 = 12n2 + -12n2

Combine like terms: 12n2 + -12n2 = 0
-54 + 11n + 6n2 = 0

Begin completing the square.  Divide all terms by
6 the coefficient of the squared term: 

Divide each side by '6'.
-9 + 1.833333333n + n2 = 0

Move the constant term to the right:

Add '9' to each side of the equation.
-9 + 1.833333333n + 9 + n2 = 0 + 9

Reorder the terms:
-9 + 9 + 1.833333333n + n2 = 0 + 9

Combine like terms: -9 + 9 = 0
0 + 1.833333333n + n2 = 0 + 9
1.833333333n + n2 = 0 + 9

Combine like terms: 0 + 9 = 9
1.833333333n + n2 = 9

The n term is 1.833333333n.  Take half its coefficient (0.9166666665).
Square it (0.8402777775) and add it to both sides.

Add '0.8402777775' to each side of the equation.
1.833333333n + 0.8402777775 + n2 = 9 + 0.8402777775

Reorder the terms:
0.8402777775 + 1.833333333n + n2 = 9 + 0.8402777775

Combine like terms: 9 + 0.8402777775 = 9.8402777775
0.8402777775 + 1.833333333n + n2 = 9.8402777775

Factor a perfect square on the left side:
(n + 0.9166666665)(n + 0.9166666665) = 9.8402777775

Calculate the square root of the right side: 3.136921704

Break this problem into two subproblems by setting 
(n + 0.9166666665) equal to 3.136921704 and -3.136921704.

Subproblem 1

n + 0.9166666665 = 3.136921704 Simplifying n + 0.9166666665 = 3.136921704 Reorder the terms: 0.9166666665 + n = 3.136921704 Solving 0.9166666665 + n = 3.136921704 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.9166666665' to each side of the equation. 0.9166666665 + -0.9166666665 + n = 3.136921704 + -0.9166666665 Combine like terms: 0.9166666665 + -0.9166666665 = 0.0000000000 0.0000000000 + n = 3.136921704 + -0.9166666665 n = 3.136921704 + -0.9166666665 Combine like terms: 3.136921704 + -0.9166666665 = 2.2202550375 n = 2.2202550375 Simplifying n = 2.2202550375

Subproblem 2

n + 0.9166666665 = -3.136921704 Simplifying n + 0.9166666665 = -3.136921704 Reorder the terms: 0.9166666665 + n = -3.136921704 Solving 0.9166666665 + n = -3.136921704 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.9166666665' to each side of the equation. 0.9166666665 + -0.9166666665 + n = -3.136921704 + -0.9166666665 Combine like terms: 0.9166666665 + -0.9166666665 = 0.0000000000 0.0000000000 + n = -3.136921704 + -0.9166666665 n = -3.136921704 + -0.9166666665 Combine like terms: -3.136921704 + -0.9166666665 = -4.0535883705 n = -4.0535883705 Simplifying n = -4.0535883705

Solution

The solution to the problem is based on the solutions from the subproblems. n = {2.2202550375, -4.0535883705}

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