6n^2-18n-13=0

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Solution for 6n^2-18n-13=0 equation:


Simplifying
6n2 + -18n + -13 = 0

Reorder the terms:
-13 + -18n + 6n2 = 0

Solving
-13 + -18n + 6n2 = 0

Solving for variable 'n'.

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

Divide each side by '6'.
-2.166666667 + -3n + n2 = 0

Move the constant term to the right:

Add '2.166666667' to each side of the equation.
-2.166666667 + -3n + 2.166666667 + n2 = 0 + 2.166666667

Reorder the terms:
-2.166666667 + 2.166666667 + -3n + n2 = 0 + 2.166666667

Combine like terms: -2.166666667 + 2.166666667 = 0.000000000
0.000000000 + -3n + n2 = 0 + 2.166666667
-3n + n2 = 0 + 2.166666667

Combine like terms: 0 + 2.166666667 = 2.166666667
-3n + n2 = 2.166666667

The n term is -3n.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3n + 2.25 + n2 = 2.166666667 + 2.25

Reorder the terms:
2.25 + -3n + n2 = 2.166666667 + 2.25

Combine like terms: 2.166666667 + 2.25 = 4.416666667
2.25 + -3n + n2 = 4.416666667

Factor a perfect square on the left side:
(n + -1.5)(n + -1.5) = 4.416666667

Calculate the square root of the right side: 2.101586702

Break this problem into two subproblems by setting 
(n + -1.5) equal to 2.101586702 and -2.101586702.

Subproblem 1

n + -1.5 = 2.101586702 Simplifying n + -1.5 = 2.101586702 Reorder the terms: -1.5 + n = 2.101586702 Solving -1.5 + n = 2.101586702 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + n = 2.101586702 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = 2.101586702 + 1.5 n = 2.101586702 + 1.5 Combine like terms: 2.101586702 + 1.5 = 3.601586702 n = 3.601586702 Simplifying n = 3.601586702

Subproblem 2

n + -1.5 = -2.101586702 Simplifying n + -1.5 = -2.101586702 Reorder the terms: -1.5 + n = -2.101586702 Solving -1.5 + n = -2.101586702 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + n = -2.101586702 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = -2.101586702 + 1.5 n = -2.101586702 + 1.5 Combine like terms: -2.101586702 + 1.5 = -0.601586702 n = -0.601586702 Simplifying n = -0.601586702

Solution

The solution to the problem is based on the solutions from the subproblems. n = {3.601586702, -0.601586702}

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