6n^2+13n-13=0

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


Simplifying
6n2 + 13n + -13 = 0

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

Solving
-13 + 13n + 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 + 2.166666667n + n2 = 0

Move the constant term to the right:

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

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

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

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

The n term is 2.166666667n.  Take half its coefficient (1.083333334).
Square it (1.173611113) and add it to both sides.

Add '1.173611113' to each side of the equation.
2.166666667n + 1.173611113 + n2 = 2.166666667 + 1.173611113

Reorder the terms:
1.173611113 + 2.166666667n + n2 = 2.166666667 + 1.173611113

Combine like terms: 2.166666667 + 1.173611113 = 3.34027778
1.173611113 + 2.166666667n + n2 = 3.34027778

Factor a perfect square on the left side:
(n + 1.083333334)(n + 1.083333334) = 3.34027778

Calculate the square root of the right side: 1.827642684

Break this problem into two subproblems by setting 
(n + 1.083333334) equal to 1.827642684 and -1.827642684.

Subproblem 1

n + 1.083333334 = 1.827642684 Simplifying n + 1.083333334 = 1.827642684 Reorder the terms: 1.083333334 + n = 1.827642684 Solving 1.083333334 + n = 1.827642684 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.083333334' to each side of the equation. 1.083333334 + -1.083333334 + n = 1.827642684 + -1.083333334 Combine like terms: 1.083333334 + -1.083333334 = 0.000000000 0.000000000 + n = 1.827642684 + -1.083333334 n = 1.827642684 + -1.083333334 Combine like terms: 1.827642684 + -1.083333334 = 0.74430935 n = 0.74430935 Simplifying n = 0.74430935

Subproblem 2

n + 1.083333334 = -1.827642684 Simplifying n + 1.083333334 = -1.827642684 Reorder the terms: 1.083333334 + n = -1.827642684 Solving 1.083333334 + n = -1.827642684 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.083333334' to each side of the equation. 1.083333334 + -1.083333334 + n = -1.827642684 + -1.083333334 Combine like terms: 1.083333334 + -1.083333334 = 0.000000000 0.000000000 + n = -1.827642684 + -1.083333334 n = -1.827642684 + -1.083333334 Combine like terms: -1.827642684 + -1.083333334 = -2.910976018 n = -2.910976018 Simplifying n = -2.910976018

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.74430935, -2.910976018}

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