18n^2-52n-81=0

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Solution for 18n^2-52n-81=0 equation:


Simplifying
18n2 + -52n + -81 = 0

Reorder the terms:
-81 + -52n + 18n2 = 0

Solving
-81 + -52n + 18n2 = 0

Solving for variable 'n'.

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

Divide each side by '18'.
-4.5 + -2.888888889n + n2 = 0

Move the constant term to the right:

Add '4.5' to each side of the equation.
-4.5 + -2.888888889n + 4.5 + n2 = 0 + 4.5

Reorder the terms:
-4.5 + 4.5 + -2.888888889n + n2 = 0 + 4.5

Combine like terms: -4.5 + 4.5 = 0.0
0.0 + -2.888888889n + n2 = 0 + 4.5
-2.888888889n + n2 = 0 + 4.5

Combine like terms: 0 + 4.5 = 4.5
-2.888888889n + n2 = 4.5

The n term is -2.888888889n.  Take half its coefficient (-1.444444445).
Square it (2.086419755) and add it to both sides.

Add '2.086419755' to each side of the equation.
-2.888888889n + 2.086419755 + n2 = 4.5 + 2.086419755

Reorder the terms:
2.086419755 + -2.888888889n + n2 = 4.5 + 2.086419755

Combine like terms: 4.5 + 2.086419755 = 6.586419755
2.086419755 + -2.888888889n + n2 = 6.586419755

Factor a perfect square on the left side:
(n + -1.444444445)(n + -1.444444445) = 6.586419755

Calculate the square root of the right side: 2.566402103

Break this problem into two subproblems by setting 
(n + -1.444444445) equal to 2.566402103 and -2.566402103.

Subproblem 1

n + -1.444444445 = 2.566402103 Simplifying n + -1.444444445 = 2.566402103 Reorder the terms: -1.444444445 + n = 2.566402103 Solving -1.444444445 + n = 2.566402103 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.444444445' to each side of the equation. -1.444444445 + 1.444444445 + n = 2.566402103 + 1.444444445 Combine like terms: -1.444444445 + 1.444444445 = 0.000000000 0.000000000 + n = 2.566402103 + 1.444444445 n = 2.566402103 + 1.444444445 Combine like terms: 2.566402103 + 1.444444445 = 4.010846548 n = 4.010846548 Simplifying n = 4.010846548

Subproblem 2

n + -1.444444445 = -2.566402103 Simplifying n + -1.444444445 = -2.566402103 Reorder the terms: -1.444444445 + n = -2.566402103 Solving -1.444444445 + n = -2.566402103 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.444444445' to each side of the equation. -1.444444445 + 1.444444445 + n = -2.566402103 + 1.444444445 Combine like terms: -1.444444445 + 1.444444445 = 0.000000000 0.000000000 + n = -2.566402103 + 1.444444445 n = -2.566402103 + 1.444444445 Combine like terms: -2.566402103 + 1.444444445 = -1.121957658 n = -1.121957658 Simplifying n = -1.121957658

Solution

The solution to the problem is based on the solutions from the subproblems. n = {4.010846548, -1.121957658}

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